# Improving the Reporting of Threats to Construct Validity

Dag I.K. Sjøberg  0000-0002-4941-7240 University of OsloNorway
<EMAIL_ADDRESS>and  Gunnar R. Bergersen  University of OsloNorway
<EMAIL_ADDRESS>
(2023)

###### Abstract.

Background: Construct validity concerns the use of indicators to measure a
concept that is not directly measurable. Aim: This study intends to identify,
categorize, assess and quantify discussions of threats to construct validity
in empirical software engineering literature and use the findings to suggest
ways to improve the reporting of construct validity issues. Method: We
analyzed 83 articles that report human-centric experiments published in five
top-tier software engineering journals from 2015 to 2019. The articles’ text
concerning threats to construct validity was divided into segments (the unit
of analysis) based on predefined categories. The segments were then evaluated
regarding whether they clearly discussed a threat and a construct. Results:
Three-fifths of the segments were associated with topics not related to
construct validity. Two-thirds of the articles discussed construct validity
without using the definition of construct validity given in the article. The
threats were clearly described in more than four-fifths of the segments, but
the construct in question was clearly described in only two-thirds of the
segments. The construct was unclear when the discussion was not related to
construct validity but to other types of validity. Conclusions: The results
show potential for improving the understanding of construct validity in
software engineering. Recommendations addressing the identified weaknesses are
given to improve the awareness and reporting of CV.

measurement, research quality, empirical research

††journalyear: 2023††copyright: rightsretained††conference: Proceedings of the
International Conference on Evaluation and Assessment in Software Engineering;
June 14–16, 2023; Oulu, Finland††booktitle: Proceedings of the International
Conference on Evaluation and Assessment in Software Engineering (EASE ’23),
June 14–16, 2023, Oulu, Finland††doi: 10.1145/3593434.3593449††isbn:
979-8-4007-0044-6/23/06††ccs: General and reference General literature

##  1\. Introduction

At least in software engineering, research is intended to improve society.
Researchers would like their research to have an impact. One might like to
demonstrate that something proposed or enhanced is better than some
alternative. In some cases, one could demonstrate improvement according to
relevant success criteria using a validated measurement instrument. For
example, less time spent could be measured using a watch. Less energy
consumption could be measured using an electronic device.

However, many success criteria and other relevant aspects are often elusive
and not directly measurable, for example, quality aspects of a software
system, such as functional suitability, reliability, usability,
maintainability and security. Such concepts cannot be measured directly; we
need a set of indicators to measure them. A concept that is not directly
measurable but is represented by indicators at the operational level to make
it measurable is called a construct. The validity of a construct, which is
called construct validity (CV), is defined by how adequate a concept
definition is and how well the indicators represent the concept (Sjøberg and
Bergersen, 2022; Cook and Campbell, 1979; Cronbach and Meehl, 1955; Ralph and
Tempero, 2018).

Several general guidelines for conducting and reporting empirical studies in
software engineering have been published (Shaw, 2003; Sjøberg et al., 2007;
Jedlitschka et al., 2008; Runeson and Höst, 2009; Kitchenham et al., 2008).
This paper is focused on managing and reporting threats to CV in particular.
We identify various weaknesses and mistakes concerning CV and propose means
for rectifying them. We address the research questions as follows:

  * •

RQ1: How are threats to CV reported in empirical software engineering?

    * –

RQ1.1: What are the topics reported in discussions about threats to CV?

    * –

RQ1.2: How is CV defined, and how is the definition reflected in the
discussion on threats to CV?

    * –

RQ1.3: Which types of threats to CV are reported?

    * –

RQ1.4: Are the threats and constructs clearly described in discussions about
threats to CV?

  * •

RQ2: How can the reporting of threats to CV be improved in light of the
answers to RQ1?

An earlier study investigated CV in a set of 83 articles that report
experiments published in five top-tier software engineering journals (Sjøberg
and Bergersen, 2022). In this paper, we analyze the same articles regarding
additional aspects of CV. We also quantify the articles on aspects already
reported in (Sjøberg and Bergersen, 2022), where the unit of analysis was
either the whole article or the sections on threats to CV. In this paper, the
unit of analysis is the segment, which can be paragraphs, sentences or
sequences of words within a sentence. This finer-grained analysis provided
deeper insights into and further quantification of CV issues.

The remainder of this paper is organized as follows. Section 2 describes the
research method, including data collection and analysis. Section 3 reports and
discusses the results. Section 4 discusses limitations of the study. Section 5
concludes.

##  2\. Research Method

###  2.1. Data Collection

This investigation uses the same set of 83 articles that provide a discussion
of CV and were analyzed in a study reported in (Sjøberg and Bergersen,
2022).111References to the 83 articles can be found in (Sjøberg and Bergersen,
2022). The articles report human-centric experiments (Kitchenham et al., 2012)
published from 2015 to 2019 in five journals with a high impact factor: IEEE
Transactions on Software Engineering (TSE), Empirical Software Engineering
(EMSE), Information and Software Technology (IST), ACM Transactions on
Software Engineering and Methodology (TOSEM), and Journal of Systems and
Software (JSS).

###  2.2. Data Analysis

Both authors simultaneously read and split the subsections or paragraphs on
“threats to construct validity” into text segments consisting of sentences or
sequences of words that describe a particular topic. We then categorized and
analyzed these segments as described below. We worked until we achieved
consensus on the segmentation, categorization and other analyses. We used the
variables as follows in our analysis:

  * •

Topics. Based on findings reported in (Sjøberg and Bergersen, 2022), we divide
the segments into the following topics:

    * –

Definition of CV. When a definition was discovered in a segment, it was
categorized into one of the following predefined options based on the
categorization used in (Sjøberg and Bergersen, 2022). CV is concerned with

      * *

how well a concept is defined and how well a set of indicators represent the
concept, cf. the definition given in Section 1,

      * *

the relationship between theory and observation,

      * *

the relationship between the goals of an experiment and its results, and

      * *

the extent to which the variables measure what they intend to measure.

      * *

A category ”miscellaneous” includes definitions of CV that we find difficult
to consider plausible. Two examples are: CV concerns how an experiment is
constructed, and CV concerns how well the treatment reflects the cause and
outcome of the effect in an experiment.

    * –

Imported construct. A segment that describes a construct that is imported from
established theories, standards or frameworks, or from articles that document
the construct.

    * –

Threats to CV. There are three types of threats to CV:

      * *

Inadequate definition of concept. It means that the definition of the concept
is unclear or ambiguous, or differs from an established understanding (Cook
and Campbell, 1979).

      * *

Underrepresentation. It means that the set of indicators used to represent a
concept is too narrow to include all important aspects or dimensions of the
concept (Messick, 1989).

      * *

Construct-representations bias. Indicators used to represent a concept are
biased if they systematically misrepresent the concept; for example, if only
one method is used to collect the data, which is called mono-method bias (Cook
and Campbell, 1979). A representation is also biased if an indicator is
outside the scope of the concept; that is, the indicator does not represent
the concept.

    * –

Internal validity. In a study that infers an effect B from a cause A, a threat
to internal validity is that there could be other causes than A for the effect
B (Shadish et al., 2002).

    * –

External validity. It concerns whether the findings of a study are
generalizable beyond the immediate study (Yin, 2003).

    * –

Statistical conclusion validity. It concerns statistical analysis of the
variables included in cause-and-effect studies.

    * –

Fundamental study challenges. A study has fundamental problems if, for
example, participants cheat, do not have sufficient time to fulfil tasks or do
not understand the material given (text may be incomprehensible due to
translation errors).

    * –

Unknown. When we cannot categorize a segment into a definition, import or any
kind of threat, we describe the topic as unknown.

  * •

Consistent use of CV definition. This binary variable indicates whether the
description of threats to CV adheres to or reflects the definition of CV given
in the same article. For example, if CV is defined as the relationship between
theory and observation, is there a reference to a theory or theoretical level,
or if the definition is whether a variable measures what it intends to
measure, is there a description of the intention?

  * •

Clarity of threat. This variable indicates how clearly a threat in a segment
is described, with the possible values: unclear, somewhat clear or clear.

  * •

Clarity of construct. This variable indicates how clearly a construct included
in a threat or another statement is described, with the possible values:
unclear, somewhat clear or clear.

##  3\. Results and Discussion

The subsections below answer research questions RQ1.1 to RQ1.4, respectively,
in the context of the sample of 83 journal articles. The last paragraph in
each subsection is a response to research question RQ2.

###  3.1. Topics in Discussions on Threats to CV

We identified 310 segments in the articles. Figure 1 shows the proportion of
topics in the collected segments. The figure also shows the proportion of text
(number of words) used to describe the various topics.

We consider it relevant to include the first three topics described in Section
2 (definition of CV, imported constructs and, of course, CV) in a section on
“Threats to construct validity”. These three topics constitute approximately
two-fifths of the total segments, meaning three-fifths include topics not
relevant to CV. The essential topic ”threats to CV” forms 27% of the segments
but is included in 60% (50) of the articles. Still, this means that 40% of the
articles do not discuss threats to CV.

The most frequent topic in the segments is internal validity, which is
discussed in 61% (51) of the articles. The aspects of internal validity
include issues such as hypothesis guessing (the participants in a study may
guess the purpose of the study), evaluation apprehension (Rosenberg, 1969)
(the participants may be worried that their potentially poor performance may
become visible to others), and experimenter expectancies (Cook and Campbell,
1979) (the researcher may intentionally or unintentionally influence the
behaviour of the participants). Such issues may impact cause-effect
relationships observed in a study and thus affect internal validity, but they
do not influence how a concept is operationalized in terms of indicators and
thus do not affect CV.

The third-largest category is the “fundamental study challenges”, which
includes a range of issues as described in Section 2. We believe that the
authors of these articles included these issues under threats to CV somewhat
arbitrarily in the absence of a common, appropriate predefined validity type.
For example, it is difficult to consider cheating and allocating too short a
time for a study as related to CV.

Most of the segments in the category ”statistical conclusion validity” discuss
measurement scales. For example, it is suggested that a finer-grained scale
might have been better than a binary scale, or the use of a Likert scale is
questioned in a given study.

Figure 1. Topics in discussions on threats to CV

The segments categorized as external validity typically discuss generalization
to other populations of people than the sample of participants or
generalizations of the study material (tasks, systems, tools, etc.). Note that
when populations of subjects or study material are not concepts or constructs
included in research questions or hypotheses, we consider them related to
external validity. If they are included as moderators in a cause-effect
relationship, the operationalization of the populations would have been
included in a discussion of CV. A further discussion of the relationship
between external validity and CV can be found in (Shadish et al., 2002).

We could not categorize approximately 5% of the segments; no threat was
indicated, nor did they describe definitions of CV or imported constructs.

Given that most of the text of sections or paragraphs entitled ”threats to CV”
in the analyzed articles are not specifically relevant to CV, it is plausible
to recommend that authors not include threats other than those to CV in such
sections and paragraphs.

###  3.2. Definition and Use of CV

Table 1. Number of articles with inconsistency between definition and use of CV 2 Definition | Articles |  Incon- sistency | %
---|---|---|---
2 Relationship between theory and observation |  15 |  14 |  93
Measure of intention |  4 |  3 |  75
Relationship between experiment goals and results |  6 |  4 |  67
Miscellaneous definitions |  8 |  3 |  38
Represents the concepts of study |  3 |  0 |  0
Total |  36 |  24 |  67

For segments that included a definition or explication of CV, we investigated
whether the use of the term CV in the subsequent segments was consistent with
the stated definition. Table 1 shows the definitions of CV and whether they
are used consistently. Two-thirds of the articles that introduce a definition
or explication of CV do not adhere to it themselves. We believe that the main
reason is that the concept of CV and many of its definitions are challenging
to understand. One indication that CV is a difficult concept is the variety of
definitions. The only definition with no inconsistencies is when CV is defined
as a set of indicators representing the concept.

The majority of the articles (57%) did not include a definition of CV. A
question is whether one needs to include a definition of CV in an empirical
paper. In the long run, CV should be sufficiently understood to avoid the need
for a definition to be included in papers that report threats to CV. However,
as shown in Table 1, as long as several definitions are in use and most of the
analyzed articles do not apply their own definition, we have not yet reached
the level of common consensus and understanding in the community.

Our recommendation may seem obvious, but in light of the results reported
above, it may still be helpful: If a definition is introduced in a paper, it
should be followed.

Figure 2. Types of threats to CV

###  3.3. Types of CV Threats

Figure 2 shows the distribution in the analyzed set of articles of the major
types of CV threats described in Section 2. Most discussions concern to what
extent a set of indicators underrepresent the concept in question. Several
articles explicitly discuss the threat of mono-operation, where only one
indicator is used. Others state that they have mitigated or even removed the
threat of underrepresentation because several indicators are used.

The discussions of bias in the representation of a concept primarily focus on
the use of only one method for data collection (mono-method) and the challenge
of using subjective data. For example, there is a threat of bias if all data
is collected using self-reporting. Generally, one would prefer objective to
subjective data. However, one might intentionally collect subjective data, for
example, if the purpose is to investigate a person’s opinions or feelings.

Note that bias in CV is not the same as bias in internal validity. In CV, it
concerns the systematic (empirical) misrepresentation of a single concept. In
internal validity, it concerns whether there are systematic problems (errors)
that prevent one from making cause-effect inferences from a study, for
example, because treatment and control groups are not equal in some aspect,
which may unintentionally skew the study outcome. Additionally, note that
”bias” in the internal validity sense may involve multiple constructs. A high
degree of CV would typically be a premise for a high degree of internal
validity; bias in CV may lead to bias in cause-effect inferences. However, one
may still have internal validity problems despite no noteworthy problems with
bias for the involved constructs.

The least discussed threat is inadequate definition of concept. One example is
an article that identifies the challenge that a tool detects a specific code
smell that was not considered a smell by developers; the operational
definition implemented in the tool did not comply with the developers’
understanding.

Given that relatively few articles discuss threats regarding how the
operationalized concepts are defined, we recommend that authors also consider
and possibly report such kinds of CV threats, in addition to
underrepresentation and bias.

Figure 3. Clarity of threats in segments by topic

###  3.4. Clarity of Threats and Constructs

One would expect that a subsection on “threats to construct validity” would
clearly describe which threats included which constructs. However, four-fifths
of the articles have at least one ambiguity regarding a threat or construct.
The severity varies; for example, an article may have some paragraphs that are
clear concerning both the threat and construct involved, whereas other
paragraphs are generally unclear. Other articles discuss a threat in detail,
but it is unclear which construct is associated with the threat or vice versa.
For example, when an article states that threats to construct validity were
mitigated because a discussion of the hypothesis of the experiment was avoided
before it was conducted, it is unclear which constructs have avoided threats.
Even though it is not explicitly stated what the threat is, we have coded this
statement as ”clear” regarding the threat because we easily understood that
the threat considered was “hypothesis guessing”. Examples of unclear threats
are descriptions of the purpose of an experiment or how a construct was
measured without any threats indicated.

Figure 4. Clarity of constructs in segments by topic

Overall, the threats are unclear in 8% and somewhat unclear in another 8% of
the segments. Segments that define CV or describe imported constructs are
exempted from this analysis. Figure 3 shows that within a given topic, only
statistical conclusion validity segments have a substantial proportion of
unclear ($4/17$ segments = 24%) or somewhat unclear (18%) threats. An example
of an unclear threat in this category is a segment that simply states that a
five-point Likert scale was used. In this case, we could categorize the topic
of the segment as statistical conclusion validity because the Likert scale was
referred to. We manage to categorize one other unclear threat to fundamental
study challenges. However, we cannot categorize the topic for most of the
segments in which the threat is unclear. Consequently, most of the segments
with unclear threats fall into the category of unknown.

Constructs were more frequently unclear than threats; the constructs were
unclear in 23% and somewhat unclear in 9% of the segments. Segments with
definitions of CV are exempted from this analysis. Three-fifths of the
articles have at least one segment with (somewhat) unclear constructs. Figure
4 shows that among the segments concerning CV, almost all constructs are clear
(only three are somewhat unclear). However, for fundamental study challenges,
there are as many unclear as clear constructs, and for internal validity, half
of the segments have unclear or somewhat unclear constructs.

Even though authors may believe that the threats and constructs in question
are apparent from the context, they may not be apparent to readers. Therefore,
we recommend that authors state threats and constructs clearly and explicitly.
Note that if it is unclear which construct(s) is in question when a threat is
described, it may not be a threat to CV but to other validity types.

##  4\. Limitations

A core limitation of our work is that we use a definition of CV on which not
everybody may agree. We adhere to a minimal definition because we believe that
such a definition is more coherent and, thus, comprehensible, which in turn is
better for the research community. Some of the threats and issues we consider
outside the notion of CV may be covered by a broader definition of CV. For
example, several of the segments that we have categorized as belonging to
internal validity were considered as belonging to CV by Cook and Campbell in
1979 (Cook and Campbell, 1979). Still, some of these types of threats, for
example, ”hypothesis guessing”, were not considered CV threats any longer in
2002 by the same authors with the addition of Shadish et al. (Shadish et al.,
2002). Shadish et al. (Shadish et al., 2002) discuss CV versus other validity
types in further detail.

The identification of topics, CV definitions and their use in CV discussions,
type of CV threats and clarity of threats and constructs is based on a
subjective assessment by the authors working together and discussing until
consensus is achieved. Although we did some automatic searches in the articles
on a few specific keywords, for example, “mono-operationalization” and “mono-
method”, the study basically faces the threat of mono-method; the single
method was the authors’ subjective assessment. Instead of working together, we
could have made the assessments independently and then reported agreement, for
example, using inter-rater reliability.

Regarding causal inferences, a plausible model would be that an unclear
understanding of what construct validity entails by one or more authors
results in a problematic article section on threats to construct validity.
However, we have not investigated the causes of our findings.

The sample we investigated comprised 83 articles that report human-centric
experiments. We make no claims of generalization of the descriptive statistics
beyond this sample. Consequently, external validity is not an issue. However,
given that the articles were published in high-quality journals, one may
conjecture that the challenges we report may be more prevalent in less
prestigious publication venues.

##  5\. Conclusion

The investigation of a sample of top-tier journal articles showed that threats
to internal, external and statistical conclusion validity and problems with
other fundamental study challenges were included under the umbrella of threats
to CV. Furthermore, among those who introduced a definition of CV, the
particular definition is not reflected in the discussion of CV for two-thirds
of the articles. Finally, one-third of the analyzed text segments were unclear
or somewhat unclear about which construct was involved in the asserted threat.
For all the cases in which the construct was unclear, none of the threats were
actually threats to CV.

To improve the awareness and reporting of CV, we provided recommendations
addressing the specific, identified weaknesses. Generally, we believe that CV
and other validity types should be more prevalent in the literature and
teaching of empirical research methods in software engineering.

## References

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# Equitable factorizations of edge-connected graphs

Morteza Hasanvand
Department of Mathematical Sciences, Sharif University of Technology, Tehran,
Iran
<EMAIL_ADDRESS>
###### Abstract

In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a
certain condition on whose degrees, can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each vertex $v\in V(G_{i})$,
$|d_{G_{i}}(v)-d_{G}(v)/k|<1$, where $1\leq i\leq k$. As application, we
deduce that every $6$-edge-connected graph $G$ can be edge-decomposed into
three factors $G_{1}$, $G_{2}$, and $G_{3}$ such that for each vertex $v\in
V(G_{i})$, $|d_{G_{i}}(v)-d_{G}(v)/3|<1$, unless $G$ has exactly one vertex
$z$ with $d_{G}(z)\stackrel{{\scriptstyle 3}}{{\not\equiv}}0$. Next, we show
that every odd-$(3k-2)$-edge-connected graph $G$ can be edge-decomposed into
$k$ factors $G_{1},\ldots,G_{k}$ such that for each vertex $v\in V(G_{i})$,
$d_{G_{i}}(v)$ and $d_{G}(v)$ have the same parity and
$|d_{G_{i}}(v)-d_{G}(v)/k|<2$, where $k$ is an odd positive integer and $1\leq
i\leq k$. Finally, we give a sufficient edge-connectivity condition for a
graph $G$ to have a parity factor $F$ with specified odd-degree vertices such
that for each vertex $v$, $|d_{F}(v)-\varepsilon d_{G}(v)|<2$, where
$\varepsilon$ is a real number with $0<\varepsilon<1$.

Keywords: Factorization; equitable edge-coloring; odd-edge-connectivity;
modulo orientation; regular graph.

##  1 Introduction

In this article, graphs may have loops and multiple edges. Let $G$ be a graph.
The vertex set, the edge set, and the maximum degree of $G$ are denoted by
$V(G)$, $E(G)$, and, $\Delta(G)$, respectively. We denote by $d_{G}(v)$ the
degree of a vertex $v$ in the graph $G$, whether $G$ is directed or not. Also
the out-degree and in-degree of $v$ in a directed graph $G$ are denoted by
$d_{G}^{+}(v)$ and $d_{G}^{-}(v)$. An orientation of $G$ is said to be
$p$-orientation, if for each vertex $v$, $d_{G}^{+}(v)\stackrel{{\scriptstyle
k}}{{\equiv}}p(v)$, where $p:V(G)\rightarrow Z_{k}$ is a mapping and $Z_{k}$
is the cyclic group of order $k$. For any two disjoint vertex sets $A$ and
$B$, we denote by $d_{G}(A,B)$ the number of edges with one end in $A$ and the
other one in $B$. Also, we denote by $d_{G}(A)$ the number of edges of $G$
with exactly one end in $A$. Note that $d_{G}(\\{v\\})$ denotes the number of
non-loop edges incident with $v$, while $d_{G}(v)$ denotes the degree of $v$.
We denote by $G[A]$ the induced subgraph of $G$ with the vertex set $A$
containing precisely those edges of $G$ whose ends lie in $A$. Let $g$ and $f$
be two integer-valued functions on $V(G)$. A parity $(g,f)$-factor of $G$
refers to a spanning subgraph $F$ such that for each vertex $v$, $g(v)\leq
d_{F}(v)\leq f(v)$, and also $d_{F}(v)$, $g(v)$, and $f(v)$ have the same
parity. An $f$-parity factor refers to a spanning subgraph $F$ such that for
each vertex $v$, $d_{F}(v)$ and $f(v)$ have the same parity. A graph $G$ is
called $m$-tree-connected, if it contains $m$ edge-disjoint spanning trees. A
graph $G$ is termed essentially $\lambda$-edge-connected, if all edges of any
edge cut of size strictly less than $\lambda$ are incident to a vertex. A
graph $G$ is called odd-$\lambda$-edge-connected, if $d_{G}(A)\geq\lambda$,
for every vertex $A$ with $d_{G}(A)$ odd. Note that $2\lambda$-edge-connected
graphs are odd-$(2\lambda+1)$-edge-connected. A factorization
$G_{1},\ldots,G_{k}$ of $G$ is called equitable factorization, if for each
vertex $v$, $|d_{G_{i}}(v)-d_{G}(v)/k|<1$, where $1\leq i\leq k$. Throughout
this article, all variables $k$ are positive and integer.

In 1971 de Werra observed that bipartite graphs admit equitable
factorizations.

###### Theorem 1.1

.([27]) Every bipartite graph $G$ can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<1.$

In 1994 Hilton and de Werra developed Theorem 1.1 to simple graphs by
replacing the following weaker version. In 2008 Hilton [13] conjectured that
for those simple graphs for which whose vertices with degree divisible by $k$
form an induced forest, the upper bound can be replaced by that of Theorem
1.1. Later, this conjecture was confirmed by Zhang and Liu (2011) [28].

###### Theorem 1.2

.([15]) Every simple graph $G$ can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|\leq 1.$

Recently, Thomassen (2019) made another attempt for developing Theorem 1.1 to
highly edge-connected regular graphs and concluded the following result. An
intresting consequence of it says that every $6$-edge-connected $3r$-regular
graph can be edge-decomposed into three $r$-regular factors.

###### Theorem 1.3

.([26]) Let $k$ and $r$ be two odd positive integers. If $G$ is an
odd-$(3k-2)$-edge-connected $kr$-regular graph, then it can be edge-decomposed
into $r$-regular factors.

In this paper, we generalize Thomassen’s result in several ways together with
developing Theorem 1.1 to highly edge-connected graphs as the next theorem. In
addition, we provide a common version for Theorem 1.3 and whose even-regular
version by replacing another concept of edge-connectivity.

###### Theorem 1.4

. Let $G$ be a $(3k-3)$-edge-connected graph, where $k$ is a positive integer.
If there is a vertex set $Z$ with $|E(G)|\stackrel{{\scriptstyle
k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$, then $G$ can be edge-decomposed into $k$
factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<1.$

In 1982 Hilton constructed the following parity version of Theorem 1.1 on the
existence of even factorizations with bounded degrees of even graphs.

###### Theorem 1.5

.([14]) Every graph $G$ with even degrees can be edge-decomposed into $k$ even
factors $G_{1},\ldots,G_{k}$ such that for each vertex $v$,
$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

In this paper, we provide two extensions of Hilton’s result in highly odd-
edge-connected graphs and highly edge-connected even graphs by proving the
following results. An interesting consequence of them says that every
$(3k-3)$-edge-connected graph of even order whose degrees lie in the set
$\\{3k,3k+2,...,5k\\}$ can be edge-decomposed into $k$ factors whose degrees
lie in the set $\\{3,5\\}$. Note that the next theorem can also be considered
as an extension of Theorem 1.3.

###### Theorem 1.6

. Let $G$ be a graph and let $k$ be an odd positive integer. If $G$ is
odd-$(3k-2)$-edge-connected, then it can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(v)$ and

$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

###### Theorem 1.7

. Let $G$ be a graph with even degrees, let $f:V(G)\rightarrow Z_{2}$ be a
mapping with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$, and
let $k$ be an even positive integer. If for every vertex set $X$ with
$\sum_{v\in X}f(v)$ odd, $d_{G}(X)\geq 3k-2$, then $G$ can be edge-decomposed
into $f$-parity factors $G_{1},\ldots,G_{k}$ such that for each $v\in
V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

In the remainder of this paper, we turn our attention to the existence of a
parity factor whose degrees are close to $\varepsilon$ of the corresponding
degrees in the main graph, and we come up with the following results on even
graphs.

###### Theorem 1.8

. Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is
a graph with even degrees, then it admits an even factor $F$ such that for
each vertex $v$,

$|d_{F}(v)-\varepsilon d_{G}(v)|<2.$

###### Theorem 1.9

. Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ positive real numbers
with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph with even
degrees, then it can be edge-decomposed into $k$ even factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$|d_{G_{i}}(v)-\varepsilon_{i}d_{G}(v)|<6.$

##  2 Preliminary results on directed graphs

The following theorem provides a relationship between orientations and
equitable factorizations of graphs which is motivated by Theorem 2 in [26].
This result plays an important role in the next section.

###### Theorem 2.1

. Every directed graph $G$ can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$ that $1\leq i\leq k$,

$|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|<1\text{ and
}|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<1.$

In particular, $|d_{G_{i}}(v)-d_{G}(v)/k|<1$, when $d^{+}_{G}(v)$ or
$d^{-}_{G}(v)$ is divisible by $k$. Furthermore, one can impose at least one
of the following conditions arbitrary:

  1. $(i)$

$|\,|E(G_{i})|-|E(G)|/k\,|<1$, where $1\leq i\leq k$.

  2. $(ii)$

If $u\in V(G)$ and $d_{G}^{+}(u)\stackrel{{\scriptstyle
k}}{{\equiv}}d_{G}^{-}(u)$, then $d_{G_{i}}(u)\stackrel{{\scriptstyle
2}}{{\equiv}}(d_{G}^{+}(u)-d_{G}^{-}(u))/k$, where $1\leq i\leq k$.

###### Proof.

The proof presented here is inspired by the proof of Theorem 2 in [26]. First
split every vertex $v$ into two vertices $v^{+}$ and $v^{-}$ such that $v^{+}$
whose incident edges were directed away from $v$ in $G$ and $v^{-}$ whose
incident edges were directed toward $v$ in $G$. In this construction, every
loop in $G$ incident with $v$ is transformed into an edge between $v^{+}$ and
$v^{-}$. Now, split every vertex in the new graph into vertices with degrees
divisible by $k$, except possibly one vertex with degree less than $k$. Call
the resulting (loopless) bipartite graph $G_{0}$. Since $G_{0}$ has maximum
degree at most $k$, it admits a proper edge-coloring with at most $k$ colors
$c_{1}\ldots,c_{k}$ (to prove this, it is enough to consider a $k$-regular
supergraph of it and apply König Theorem [19]). Let $G_{i}$ be the factor of
$G$ consisting of the edges of $G$ corresponding to the edges with color
$c_{i}$ in $G_{0}$. Note that for every vertex $v$, at least $\lfloor
d^{+}_{G}(v)/k\rfloor$ (resp. $\lfloor d^{-}_{G}(v)/k\rfloor$) edges with
color $c_{i}$ are incident with the vertex $v^{+}$ (resp. $v^{-}$) in $G_{0}$.
Moreover, at most $\lceil d^{+}_{G}(v)/k\rceil$ (resp. $\lceil
d^{-}_{G}(v)/k\rceil$) edges with color $c_{i}$ are incident with the vertex
$v^{+}$ (resp. $v^{-}$) in $G_{0}$. This completes the proof of the first
part.

First we are going to prove item (i). Let us consider such a proper edge-
coloring with the minimum $\sum_{1\leq i\leq k}|m_{i}-m/k|$, where
$m_{i}=|E(G_{i})|$ and $m=|E(G)|$. We claim that $|m_{i}-m/k|<1$ for every
color $c_{i}$. Suppose, to the contrary, that there is a color $c_{i}$ with
$|m_{i}-m/k|\geq 1$. We may assume that $m_{i}\geq m/k+1$; as the proof of the
case $m_{i}\leq m/k-1$ is similar. Thus there is another color $c_{j}$ with
$c_{j}<m/k$ so that $m_{i}>m_{j}+1$. Let $H$ be the factor of $G_{0}$
consisting of the edges that are colored with $c_{i}$ or $c_{j}$. According to
the proper edge coloring of $G_{0}$, the graph $H$ must be the union of some
paths and cycles. Since all cycles have even size and $m_{i}>m_{j}$, it is
easy to check that there is a path $P$ such that whose edges are colored
alternatively by $c_{i}$ and $c_{j}$, and also whose end edges are colored
with $c_{i}$. Now, it is enough to exchange the colors of the edges of $P$ to
find another proper edge-coloring satisfying $m^{\prime}_{i}=m_{i}-1$ and
$m^{\prime}_{j}=m_{j}+1$ where $m^{\prime}_{i}$ and $m^{\prime}_{j}$ are the
number of edges having these new colors. Since
$|m^{\prime}_{j}-m/k|+|m^{\prime}_{i}-m/k|<(|m_{j}-m/k|+1)+(|m_{i}-m/k|-1)$,
one can easily derive at a contradiction. Therefore, $G_{1},\ldots,G_{k}$ are
the desired factors we are looking for.

Now, we are going to prove item (ii). Let us add some artificial edges to the
graph $G_{0}$. For every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle
k}}{{\equiv}}d_{G}^{-}(u)\stackrel{{\scriptstyle k}}{{\not\equiv}}0$, denote
by $u_{0}^{+}$ and $u_{0}^{-}$ the two vertices in $G_{0}$ having the same
degree strictly less than $k$ obtained from splitting of $u^{+}$ and $u^{-}$.
Add some new artificial edges between these two vertices to construct these
vertices with degree $k$. Apply this method for all such vertices $u$
described above. Consider a proper edge-coloring with at most $k$ colors
$c_{1}\ldots,c_{k}$ for the new resulting bipartite graph $G^{\prime}_{0}$.
Similarly, we define $G_{i}$ to be the factor of $G$ consisting of the edges
of $G$ corresponding to the edges with color $c_{i}$ in $G_{0}$. Note that for
every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle
k}}{{\equiv}}d_{G}^{-}(u)\stackrel{{\scriptstyle k}}{{\not\equiv}}0$, either
one edge between $u_{0}^{+}$ and $u_{0}^{-}$ in $G^{\prime}_{0}$ (may be an
artificial edge) is colored with $c_{i}$ or two edges incident with
$u_{0}^{+}$ and $u_{0}^{-}$ in $G_{0}$ are colored with $c_{i}$. Thus for
every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle
k}}{{\equiv}}d_{G}^{-}(u)$, we must have
$d^{-}_{G_{i}}(u)-d^{-}_{G}(u)/k=d^{+}_{G_{i}}(u)-d^{+}_{G}(u)/k$ and so
$d_{G_{i}}(u)\stackrel{{\scriptstyle
2}}{{\equiv}}d^{+}_{G_{i}}(u)-d^{-}_{G_{i}}(u)=(d_{G}^{+}(u)-d_{G}^{-}(u))/k$.
Therefore, $G_{1},\ldots,G_{k}$ are again the desired factors we are looking
for.  $\Box$

###### Corollary 2.2

.([2]) Every graph $G$ can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each factor $G_{i}$,
$|\,|E(G_{i})|-|E(G)|/k\,|<1$, and for each $v\in V(G_{i})$,

$\lfloor\frac{d_{G}(v)}{2k}\rfloor+\lfloor\frac{d_{G}(v)+1}{2k}\rfloor\leq
d_{G_{i}}(v)\leq\lceil\frac{d_{G}(v)}{2k}\rceil+\lceil\frac{d_{G}(v)-1}{2k}\rceil.$

###### Proof.

First, consider an orientation for $G$ such that for each vertex $v$,
$|d^{+}_{G}(v)-d^{-}_{G}(v)|\leq 1$. Next, apply Theorem 2.1 (i) to this
directed graph.  $\Box$

##  3 Factorizations of edge-connected graphs

In this section, we are going to give a sufficient condition for the existence
of equitable factorizations in highly edge-connected graphs. For this purpose,
let us recall the following lemma from [12, 22].

###### Lemma 3.1

.([12, 22]) Let $G$ be a graph, let $k$ be an integer, $k\geq 2$, and let
$p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle
k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(3k-3)$-edge-connected or
$(2k-2)$-tree-connected, then it admits a $p$-orientation modulo $k$.
Furthermore, the result holds for loopless $(2k-1)$-edge-connected essentially
$(3k-3)$-edge-connected graphs provided that $p(v)=0$ or
$p(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(v)$ for each vertex $v$.

Before stating the main result, we begin with the following theorem that only
needs replacing a weaker condition on a single arbitrary vertex.

###### Theorem 3.2

. Let $G$ be a graph with $z\in V(G)$ and let $k$ be a positive integer. If
$G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it can be
edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ satisfying
$||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<\begin{cases}2&\text{when $v=z$};\\\
1,&\text{otherwise}.\end{cases}$

Furthermore, for the vertex $z$, we can also have
$|d_{G_{i}}(z)-d_{G_{j}}(z)|\leq 2$.

###### Proof.

According to Lemma 3.1, the graph $G$ admits an orientation such that for each
$v\in V(G)\setminus z$, $d^{+}_{G}(v)\stackrel{{\scriptstyle k}}{{\equiv}}0$,
and $d^{+}_{G}(z)\stackrel{{\scriptstyle k}}{{\equiv}}|E(G)|$. Now, it is
enough to apply Theorem 2.1 (i).  $\Box$

Now, we are reedy to state the main result of this section which is a
strengthened version of Theorem 1.4.

###### Theorem 3.3

. Let $G$ be a graph and let $k$ be a positive integer. Assume that $G$ is
$(3k-3)$-edge-connected or $(2k-2)$-tree-connected. If there is a vertex set
$Z$ with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$,
then $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$
satisfying $||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<1.$

Furthermore, the result holds for loopless $(2k-1)$-edge-connected essentially
$(3k-3)$-edge-connected graphs.

###### Proof.

For each $v\in Z$, define $p(v)=d_{G}(v)$, and for each $v\in V(G)\setminus
Z$, define $p(v)=0$. By the assumption, we have $|E(G)|\stackrel{{\scriptstyle
k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Thus by Lemma 3.1, the graph $G$ admits a
$p$-orientation modulo $k$ so that for each $v\in Z$, $d^{-}_{G}(v)$ is
visible by $k$ and for each $v\in V(G)\setminus Z$, $d^{+}_{G}(v)$ is
divisible by $k$. Hence the assertion follows from Theorem 2.1 (i).  $\Box$

Graphs with size divisible by $k$ are natural candidates for graphs satisfying
the assumptions of Theorem 3.3. We examine them to deduce the following
corollary.

###### Corollary 3.4

. Let $G$ be a graph of size divisible by $k$. If $G$ is $(3k-3)$-edge-
connected or $(2k-2)$-tree-connected, then it can be edge-decomposed into $k$
factors $G_{1},\ldots,G_{k}$ with the same size such that for each $v\in
V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<1.$

###### Proof.

Apply Theorem 3.3 with $Z=\emptyset$. Note that $|E(G)|$ is divisible by $k$.
$\Box$

The next corollary gives a criterion for the existence of a $3$-equitable
factorization in edge-connected graphs.

###### Corollary 3.5

. Let $G$ be a $6$-edge-connected graph. Then $G$ has not exactly one vertex
$z$ with $d_{G}(z)\stackrel{{\scriptstyle 3}}{{\not\equiv}}0$ if and only if
it can be edge-decomposed into three factors $G_{1}$, $G_{2}$, and $G_{3}$
such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/3|<1.$

Furthermore, the result holds for loopless $5$-edge-connected essentially
$6$-edge-connected graphs.

###### Proof.

Set $Z_{0}$ to be the set of all vertices of $G$ such that whose degrees are
not divisible by $3$. First assume that $|Z_{0}|\neq 1$. If $|Z_{0}|=0$, then
$2|E(G)|$ must be divisible by $3$ and so $|E(G)|$ must be divisible by $3$.
Therefore, it is easy to check that there is a subset $Z$ of $Z_{0}$ such that
$|E(G)|\stackrel{{\scriptstyle 3}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$ whether
$|Z_{0}|\geq 2$ or not. Hence by Theorem 3.3, the graph $G$ has the desired
factorization. Now, assume $Z_{0}=\\{z\\}$. Suppose, to the contrary, that $G$
can be edge-decomposed into three factors $G_{1}$, $G_{2}$, and $G_{3}$ such
that for each $v\in V(G_{i})$, $|d_{G_{i}}(v)-d_{G}(v)/3|<1$. We may therefore
assume that $d_{G_{1}}(z)=\lfloor d_{G}(z)/3\rfloor$ and $d_{G_{2}}(z)=\lceil
d_{G}(z)/3\rceil$. On the other hand, for all vertices $v\in
V(G)\setminus\\{z\\}$, $d_{G_{1}}(v)=d_{G_{2}}(v)=d_{G}(v)/3$ which implies
that $d_{G_{1}}(z)$ and $d_{G_{2}}(z)$ have the same parity, because of the
handshaking lemma. This is contradiction, as desired.  $\Box$

As we observed above, graphs with a number of vertices whose degrees are not
divisible by $k$ are other natural candidates for graphs satisfying the
assumptions of Theorem 3.3. By employing the following lemma, we examine a
special case of them to imply the next corollary.

###### Lemma 3.6

.(Chowla [6]) Let $k$ be an integer number with $k\geq 2$ and let
$x_{1},\ldots,x_{k-1}\in Z_{k}\setminus\\{0\\}$ be $k-1$ integer numbers
coprime with $k$ (not necessarily distinct). If $m\in Z_{k}$, then there is a
subset $Z\subseteq\\{1,\ldots,k-1\\}$ such that $m\stackrel{{\scriptstyle
k}}{{\equiv}}\sum_{i\in Z}x_{i}$.

###### Corollary 3.7

. Let $G$ be a $(3k-3)$-edge-connected graph where $k$ is a prime number. If
$G$ contains at least $k-1$ vertices whose degree are coprime with $k$, then
it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ satisfying
$||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$,
$|d_{G_{i}}(v)-d_{G}(v)/k|<1.$

###### Proof.

Let $Z_{0}$ be the set of all vertices of $G$ such that whose degrees are
coprime with $k$. Since $|Z_{0}|\geq k-1$, there is a subset $Z$ of $Z_{0}$
such that $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$
according to Lemma 3.6. Hence the assertion follows from Theorem 3.3
immediately.  $\Box$

It is perhaps surprising that the lower bound of $k-1$ in the above-mentioned
corollary is best possible according to the following observation. It remains
to decide whether the statement of Corollary 3.7 holds if $G$ contains at
least $k-1$ vertices whose degree are not divisible by $k$. The statement is
obviously true for all prime numbers $k$.

###### Observation 3.8

. For every integer $k$ with $k\geq 2$, there are infinitely many highly edge-
connected graphs $G$ having at least $k-2$ vertices whose degrees are coprime
with $k$, while $G$ cannot be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$|d_{G_{i}}(v)-d_{G}(v)/k|<1$.

###### Proof.

Let $r$ be an arbitrary odd positive integer. Choose a $kr$-edge-connected
graph $G$ of odd order in which whose degrees are $rk$ except for $k-2$
vertices having degree $kr+1$ (for $k$ even, one can consider a $kr$-edge-
connected $kr$-regular large graph of odd order and insert a new perfect
matching of size $k/2-1$ to it). If $G$ has the desired factorization, then
according to the vertex degree, there must be a factor $G_{i}$ such that for
all vertices $v$, $d_{G_{i}}(v)=\lfloor d_{G}(v)/k\rfloor$. This implies that
$G$ an $r$-regular factor which is not possible, because of the handshaking
lemma. Hence the assertion holds.  $\Box$

##  4 Parity factorizations of odd-edge-connected graphs

In this section, we are going to develop each of Theorems 1.3 and 5.13 in two
ways based on Theorem 2.1. For this purpose, we first need the following lemma
which improves the edge-connectivity needed in Lemma 3.1 for a special case.

###### Lemma 4.1

.([22]) Let $k$ be an odd positive integer. If $G$ is an odd-$(3k-2)$-edge-
connected graph, then it admits an orientation such that for each vertex $v$,
$d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(v)$.

The following theorem generalizes Theorem 5.13 to non-Eulerian graphs for the
case that $k$ is odd. This result can also be considered as an improvement of
a result due to Shu, Zhang, and Zhang (2012) [24] who proved this result for
odd-$(2k-1)$-edge-connected graphs without considering the restriction on
vertex degrees.

###### Theorem 4.2

. Let $k$ be an odd positive integer. If $G$ is an odd-$(3k-2)$-edge-connected
or $(2k-2)$-tree-connected graph, then it can be edge-decomposed into $k$
factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(v)$ and

$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

###### Proof.

According to Lemmas 3.1 and 4.1, the edge-connectivity condition implies the
graph $G$ admits an orientation such that for each vertex $v$,
$d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(v)$. By Theorem
2.1 (ii), the graph $G$ can be edge-decomposed into $k$ factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<1$ and
$|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|<1$, and also
$d_{G_{i}}(v)\stackrel{{\scriptstyle
2}}{{\equiv}}(d_{G}^{+}(v)-d_{G}^{-}(v))/k$. Therefore,
$|d_{G_{i}}(v)-d_{G}(v)/k|\leq|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|+|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<2$,
and all $d_{G_{i}}(v)$ have the same parity for which $1\leq i\leq k$. Since
$d_{G}(v)=\sum_{1\leq j\leq k}d_{G_{j}}(v)$ and $k$ is odd, $d_{G_{i}}(v)$ and
$d_{G}(v)$ must have the same parity. Hence the theorem holds.  $\Box$

The edge-connectivity needed in Lemma 3.1 can also be improved for the
following special case. We are going to apply it to deduce the next result on
Eulerian graphs.

###### Lemma 4.3

.([12]) Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and
let $Q\subseteq V(G)$ with $|Q|$ even. If $d_{G}(X)\geq 3k-2$ for every
$X\subseteq V(G)$ with $|X\cap Q|$ odd, then $G$ admits an orientation such
that for each $v\in V(G)\setminus Q$, $d_{G}^{+}(v)=d_{G}^{-}(v)$, and for
each $v\in Q$, $|d_{G}^{+}(v)-d_{G}^{-}(v)|=k$.

The next theorem strengthens Theorem 5.13 by imposing a new parity restriction
on degrees of the desire factors for graphs with higher edge-connectivity.

###### Theorem 4.4

. Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let
$f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)$ even. If
$d_{G}(X)\geq 3k-2$ for every vertex set $X$ with $\sum_{v\in X}f(v)$ odd, or
$G$ is $(2k-2)$-tree-connected, then $G$ can be edge-decomposed into
$f$-parity factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

###### Proof.

According to Lemmas 3.1 and 4.3, the edge-connectivity condition implies the
graph $G$ admits an orientation such that for each vertex $v$,
$d_{G}^{+}(v)=d_{G}^{-}(v)$ when $f(v)$ is even, and
$|d_{G}^{+}(v)-d_{G}^{-}(v)|=k$ when $f(v)$ is odd. Thus by Theorem 2.1 (ii),
the graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$
such that for each $v\in V(G_{i})$, $|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<1$ and
$|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|<1$, and also
$d_{G_{i}}(v)\stackrel{{\scriptstyle
2}}{{\equiv}}(d_{G}^{+}(v)-d_{G}^{-}(v))/k$. Therefore,
$|d_{G_{i}}(v)-d_{G}(v)/k|\leq|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|+|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<2$
and $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f(v)$. Hence the theorem
holds.  $\Box$

An attractive application of Theorems 4.2 and 4.4 is given in the following
corollary.

###### Corollary 4.5

. Let $k$ and $r$ be two positive integers. Let $G$ be a graph whose degrees
lie in the set $\\{rk,rk+2,...,rk+2k\\}$ for which $r|V(G)|$ is even. If $G$
is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it can be edge-
decomposed into $k$ $\\{r,r+2\\}$-factors.

###### Proof.

If $k$ is odd, then by Theorem 4.2, the graph $G$ can be edge-decomposed into
$k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,
$d_{G_{i}}(v)\stackrel{{\scriptstyle
2}}{{\equiv}}d_{G}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}r$ and $r-2\leq
d_{G}(v)/k-2<d_{G_{i}}(v)<d_{G}(v)/k+2\leq r+4.$ If $k$ is even, then by
applying Theorem 4.4 with $f(v)=r$, these factors can similarly be found such
that for each $v\in V(G_{i})$, $d_{G_{i}}(v)\stackrel{{\scriptstyle
2}}{{\equiv}}r$ and $r-2<d_{G_{i}}(v)<r+4.$ This completes the proof.  $\Box$

###  4.1 Graphs with degrees divisible by $k$: regular factorizations

In this subsection, we restrict out attention to graphs with degrees divisible
by $k$ and derive some results based on the following reformulation of
Theorems 4.2 and 4.4 on this family of graphs.

###### Theorem 4.6

. Let $k$ be a positive integer and let $G$ be a graph with size and degrees
divisible by $k$. Take $Q$ to be the set of all vertices $v$ with $d_{G}(v)/k$
odd. If for every vertex X with $|X\cap Q|$ odd,

$d_{G}(X)\geq 3k-2,$

then $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such
that for each $v\in V(G_{i})$, $d_{G_{i}}(v)=d_{G}(v)/k$.

###### Proof.

To show a a directive proof, we first consider an orientation for $G$ such
that out-degree of each vertex is divisible by $k$ using Lemmas 4.1 and 4.3.
Next, it is enough to apply Theorem 2.1 (i).  $\Box$

The following corollary partially confirms Conjecture 2 in [26] by giving a
supplement for Theorem 1.3.

###### Corollary 4.7

. Let $r$ be an odd positive integer. If $G$ is a $kr$-regular graph $G$ of
even order satisfying $d_{G}(A)\geq 3k-2$, for every vertex set $A$ with $|A|$
odd, then it can be edge-decomposed into $r$-factors.

###### Proof.

Since $G$ has even order, its size must be divisible by $k$. Note also that
for each vertex $v$, $d_{G}(v)/k$ is odd. Thus the assertion follows from
Theorem 4.6 immediately,  $\Box$

The following corollary gives a supplement for Theorem 3 in [26].

###### Corollary 4.8

. Let $r$ be a positive integer and let $r_{1},\ldots,r_{m}$ be $m$ positive
integers satisfying $r=r_{1}+\cdots+r_{m}$ and $r_{i}\geq r/k-1\geq 2$ in
which $k$ is positive divisor of $r$ with $r/k$ odd. If $G$ is an $r$-regular
graph of even order and for every vertex $X$ with $|X|$ odd,

$d_{G}(X)\geq 3k-2,$

then it can be edge-decomposed into factors $G_{1},\ldots,G_{m}$ such that
every graph $G_{i}$ is $r_{i}$-regular.

###### Proof.

Apply Corollary 4.7 along with a similar argument stated in the proof of
Theorem 3 in [26].  $\Box$

The following corollary is a counterpart of Corollary 4.8 and replaces a
weaker edge-connectivity condition compared to Theorem 4 in [26]. The proof
technique shows a worthwhile application of this kind of odd-edge-connectivity
for working with supergraphs.

###### Corollary 4.9

. Let $r$ be a positive integer and let $r_{1},\ldots,r_{m}$ be $m$ positive
integers satisfying $r=r_{1}+\cdots+r_{m}$ and $r_{i}\geq r/k-1\geq 2$ in
which $k$ is positive divisor of $r$ with $r/k$ odd. If $G$ is a graph with
$r|V(G)|$ even satisfying $\Delta(G)\leq r$ and for every vertex X with $|X|$
odd,

$d_{G}(X)\geq 3k-3,$

then it can be edge-decomposed into factors $G_{1},\ldots,G_{m}$ satisfying
$\Delta(G_{i})\leq r_{i}$ for each $i$ with $1\leq i\leq m$.

###### Proof.

Add some edges to $G$, as long as possible, such that the resulting graph
$G^{\prime}$ has maximum degree at most $r$. Since adding loops is possible
and $r|V(G)|$ is even, the graph $G^{\prime}$ must be $r$-regular. Obviously,
for every vertex set $X$ with $|X|$ odd, we still have $d_{G^{\prime}}(X)\geq
3k-3$. In addition, since $d_{G^{\prime}}(X)$ and $k$ have the same parity, it
implies that $d_{G^{\prime}}(X)\geq 3k-2$. Thus by Corollary 4.8, the graph
$G^{\prime}$ can be edge-decomposed into factors
$G^{\prime}_{1},\ldots,G^{\prime}_{m}$ such that every graph $G^{\prime}_{i}$
is $r_{i}$-regular. Now, it is enough to induce this factorization for $G$ to
complete the proof.  $\Box$

##  5 A sufficient edge-connectivity condition for the existence of a parity
factor

In 1956 Hoffman made the following theorem on the the existence factors in
bipartite graphs whose degrees are close to $\varepsilon$ of the corresponding
degrees in the main graph.

###### Theorem 5.1

.([16]) Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If
$G$ is a bipartite graph, then it has a factor $F$ such that for each vertex
$v$, $|d_{F}(v)-\varepsilon d_{G}(v)|<1.$

In 1983 Kano and Saito generlized Theorem 5.1 to general graphs as the next
version.

###### Theorem 5.2

.([18]) Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If
$G$ is a graph, then it has a factor $F$ such that for each vertex $v$,
$|d_{F}(v)-\varepsilon d_{G}(v)|\leq 1.$

In 2007 Correa and Goemans [8] formulated the following factorization version
of Theorem 5.1 for bipartite graphs. Later, Correa and Matamala (2008) [7]
remarked that it is possible to generalize their result to general graphs by
replacing Theorem 5.2 in their proof, and Feige and Singh (2008) [10]
introduced an interesting alternative proof for this theorem using linear
algebraic techniques.

###### Theorem 5.3

.([7, 8]) Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ nonnegative real
numbers with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph,
then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that
for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-\varepsilon_{i}d_{G}(v)|<3.$

In this section, we shall prove the parity versions of Theorems 5.2 and 5.3
which were mentioned in the introduction as Theorems 1.8 and 1.9. Besides
them, we also generalize a recent result in [12]. Our proofs are based on the
following well-known lemma due to Lovász (1972).

###### Lemma 5.4

.([21]) Let $G$ be a graph and let $g$ and $f$ be two integer-valued functions
on $V(G)$ satisfying $g(v)\leq f(v)$ and $g(v)\stackrel{{\scriptstyle
2}}{{\equiv}}f(v)$ for each $v\in V(G)$. Then $G$ has a parity $(g,f)$-factor
if and only if for all disjoint subsets $A$ and $B$ of $V(G)$,

$\omega_{f}(G,A,B)<2+\sum_{v\in A}f(v)+\sum_{v\in
B}(d_{G}(v)-g(v))-d_{G}(A,B),$

where $\omega_{f}(G,A,B)$ denotes the number of components $G[X]$ of
$G\setminus(A\cup B)$ satisfying $\sum_{v\in X}f(v)\stackrel{{\scriptstyle
2}}{{\not\equiv}}d_{G}(X,B)$.

###  5.1 Parity $\varepsilon$-factors

The following theorem provides a parity version for Theorem 5.2 on edge-
connected graphs. Note that some edge-connected versions of that theorem were
former studied in [3, 9, 17].

###### Theorem 5.5

. Let $G$ be a connected graph, let $f:V(G)\rightarrow Z_{2}$ be a mapping
with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$, and let
$\varepsilon$ be a real number with $0<\varepsilon<1$. If for every nonempty
proper subset $X$ of $V(G)$,

$d_{G}(X)\geq\begin{cases}1/\varepsilon,&\text{when $\sum_{v\in X}f(v)$ is
odd};\\\ 1/(1-\varepsilon),&\text{when $\sum_{v\in X}(d_{G}(v)-f(v))$ is
odd},\end{cases}$

then $G$ has an $f$-parity factor $F$ such that for each vertex $v$,

$\lfloor\varepsilon d_{G}(v)\rfloor-1\leq d_{F}(v)\leq\lceil\varepsilon
d_{G}(v)\rceil+1.$

Furthermore, for a given arbitrary vertex $z$, we can arbitrary have
$d_{F}(z)\geq\varepsilon d_{G}(z)$ or $d_{F}(z)\leq\varepsilon d_{G}(z)$.

###### Proof.

For each vertex $v$, let us define $g_{0}(v)\in\\{\lfloor\varepsilon
d_{G}(v)\rfloor-1,\lfloor\varepsilon d_{G}(v)\rfloor\\}$ and
$f_{0}(v)\in\\{\lceil\varepsilon d_{G}(v)\rceil,\lceil\varepsilon
d_{G}(v)\rceil+1\\}$ such that $g_{0}(v)\stackrel{{\scriptstyle
2}}{{\equiv}}f_{0}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f(v)$. If
$g(z)<\varepsilon d_{G}(z)$ and our goal is to impose that
$d_{F}(z)\geq\varepsilon d_{G}(z)$, we will replace $g_{0}(z)$ by
$g_{0}(z)+2$, and if $f(z)>\varepsilon d_{G}(z)$ and our goal is to impose
that $d_{F}(z)\leq\varepsilon d_{G}(z)$, we will replace $f_{0}(z)$ by
$f_{0}(z)-2$. Let $A$ and $B$ be two disjoint vertex subsets of $V(G)$ with
$A\cup B\neq\emptyset$. By the definition of $g$ and $f$, we must have

${\sum_{v\in A}\varepsilon d_{G}(v)+\sum_{v\in
B}(1-\varepsilon)d_{G}(v)<2+\sum_{v\in A}f_{0}(v)+\sum_{v\in
B}(d_{G}(v)-g_{0}(v)),}$ (1)

whether $z\in A\cup B$ or not. Take $P$ to be the collection of all vertex
sets $X$ such that $G[X]$ is a component of $G\setminus(A\cup B)$ satisfying
$\sum_{v\in X}f(v)\stackrel{{\scriptstyle 2}}{{\not\equiv}}d_{G}(X,B)$. It is
easy to check that

${\sum_{X\in P}(\varepsilon
d_{G}(X,A)+(1-\varepsilon)d_{G}(X,B))\leq\sum_{v\in A}\varepsilon
d_{G}(v)+\sum_{v\in B}(1-\varepsilon)d_{G}(v)\,-d_{G}(A,B).}$ (2)

Define $P_{c}=\\{X\in P:d_{G}(X,A)>0\text{ and }d_{G}(X,B)>0\\}$. Obviously,

$|P_{c}|=\sum_{X\in P_{c}}(\varepsilon+(1-\varepsilon))\leq\sum_{X\in
P_{c}}(\varepsilon d_{G}(X,A)+(1-\varepsilon)d_{G}(X,B)).$

Set $P_{a}=\\{X\in P:d_{G}(X,A)=0\\}$ and $P_{b}=\\{X\in P:d_{G}(X,B)=0\\}$.
According to the definition, for every $X\in P_{a}$, we have

$\sum_{v\in X}f(v)\stackrel{{\scriptstyle
2}}{{\not\equiv}}d_{G}(X,B)=d_{G}(X)\stackrel{{\scriptstyle
2}}{{\equiv}}\sum_{v\in X}d_{G}(v),$

which implies that $\sum_{v\in X}(d_{G}(v)-f(v))$ is odd. Similarly, for every
$X\in P_{b}$, $\sum_{v\in X}f(v)$ must be odd. Thus by the assumption, we must
have

$|P_{a}|\leq\sum_{X\in P_{a}}(1-\varepsilon)d_{G}(X)=\sum_{X\in
P_{a}}(\varepsilon d_{G}(X,A)+(1-\varepsilon)d_{G}(X,B)),$

and also

$|P_{b}|\leq\sum_{X\in P_{b}}\varepsilon d_{G}(X)=\sum_{X\in
P_{b}}(\varepsilon d_{G}(X,A)+(1-\varepsilon)d_{G}(X,B)).$

Therefore,

${|P|=|P_{a}|+|P_{b}|+|P_{c}|\leq\sum_{X\in P}(\varepsilon
d_{G}(X,A)+(1-\varepsilon)d_{G}(X,B)).}$ (3)

According to Relations (1), (2), and (3), one can conclude that

$\omega_{f}(G,A,B)=|P|<2+\sum_{v\in A}f_{0}(v)+\sum_{v\in
B}(d_{G}(v)-g_{0}(v))-d_{G}(A,B).$

When both of the sets $A$ and $B$ are empty, the above-mentioned inequality
must automatically hold, because $\omega_{f}(G,\emptyset,\emptyset)=0$. Thus
the assertion follows from Lemma 5.4.  $\Box$

###### Remark 5.6

. Note that if we had replaced the weaker condition $|d_{F}(z)-\varepsilon
d_{G}(z)|\leq 1$ for the vertex $z$, we could impose this condition for
another vertex as well. In particular, we could impose this condition for
three arbitrary vertices provided that $\varepsilon=1/2$.

The following corollary is an improved version of the main result in [5] by
replacing an odd edge-connectivity condition. Note that $r$-regular bipartite
graphs are in the class of odd-$r$-edge-connected graphs.

###### Corollary 5.7

.([4, 5]) Let $r$ and and $r_{0}$ be two odd positive integers with $r\geq
r_{0}$. If $G$ is an odd-$\lceil r/r_{0}\rceil$-edge-connected $r$-regular
graph, then it contains an $r_{0}$-factor.

###### Proof.

Apply Theorem 5.5 with $\varepsilon=r_{0}/r$ and $f(v)=r_{0}$ (mod $2$).
$\Box$

###### Corollary 5.8

.([12]) Let $G$ be a graph and let $f:V(G)\rightarrow Z_{2}$ be a mapping with
$\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$. If $G$ is
$2$-edge-connected, then it has an $f$-parity factor $F$ such that for each
vertex $v$,

$\lfloor\frac{d_{G}(v)}{2}\rfloor-1\leq
d_{F}(v)\leq\lceil\frac{d_{G}(v)}{2}\rceil+1.$

Furthermore, for a given arbitrary vertex $z$, we can arbitrary have
$d_{F}(z)\geq d_{G}(z)/2$ or $d_{F}(z)\leq d_{G}(z)/2$.

###### Proof.

Apply Theorem 5.5 with $\varepsilon=1/2$.  $\Box$

###### Corollary 5.9

. Let $G$ be a graph and let $f:V(G)\rightarrow Z_{2}$ be a mapping with
$\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$. If $G$ is
$3$-edge-connected, then it has an $f$-parity factor $F$ such that for each
vertex $v$,

$\lfloor\frac{2\,d_{G}(v)}{3}\rfloor-1\leq
d_{F}(v)\leq\lceil\frac{2\,d_{G}(v)}{3}\rceil+1.$

###### Proof.

Apply Theorem 5.5 with $\varepsilon=2/3$.  $\Box$

An immediate consequence of the following corollary was appeared in [20] which
says that every $2$-edge-connected graph with minimum degree at least three
admits a factor whose degrees are positive and even.

###### Corollary 5.10

. Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 2/3$. If $G$
is a $2$-edge-connected graph, then it has an even factor $F$ such that for
each vertex $v$,

$|d_{F}(v)-\varepsilon d_{G}(v)|<2.$

The next corollary provides an improved version for a result in [23] due to
Lu, Wang, and Lin (2015) who proved that every $2m$-edge-connected graph with
minimum degree at least $2m+1$ contains an even factor with minimum degree at
least $2m$.

###### Corollary 5.11

. Let $G$ be a $2$-edge-connected graph and let $\varepsilon$ be a real number
with $2/3\leq\varepsilon<1$. If $G$ is odd-$\lceil
1/(1-\varepsilon)\rceil$-edge-connected, then it has an even factor $F$ such
that for each vertex $v$,

$|d_{F}(v)-\varepsilon d_{G}(v)|<2.$

When the main graph has no odd edge-cuts, one can derive the following simpler
version of Corollaries 5.10 and 5.11.

###### Corollary 5.12

. Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is
a graph with even degrees, then it admits an even factor $F$ such that for
each vertex $v$,

$|d_{F}(v)-\varepsilon d_{G}(v)|<2.$

Here, we introduce a simple inductive proof for Theorem 8 in [14] based on
Corollary 5.12.

###### Corollary 5.13

.([14]) Every graph $G$ with even degrees can be edge-decomposed into $k$ even
factors $G_{1},\ldots,G_{k}$ such that for each vertex $v$,
$|d_{G_{i}}(v)-d_{G}(v)/k|<2.$

###### Proof.

By induction on $k$. We may assume that $k\geq 2$ as the assertion trivially
holds when $k=1$. According to Corollary 5.12, the graph can be edge-
decomposed into two even factors $G^{\prime}$ and $G_{k}$ such that for each
vertex $v$, $|d_{G_{k}}(v)-\frac{1}{k}d_{G}(v)|<2$. By the induction
hypothesis, the graph $G^{\prime}$ can be edge-decomposed into $k-1$ even
factors $G_{1},\ldots,G_{k-1}$ such that for each vertex $v$,
$|d_{G_{i}}(v)-\frac{1}{k-1}d_{G^{\prime}}(v)|<2.$ We claim that these are the
desired factors we are looking for. Let $v\in V(G)$. Since
$d_{G^{\prime}}(v)\leq\frac{k-1}{k}d_{G}(v)+2-\frac{1}{k}$, we must have
$d_{G_{i}}(v)\leq\frac{1}{k-1}(d_{G^{\prime}}(v)-1)+2\leq\frac{1}{k}(d_{G}(v)+1)+2.$
Since $d_{G}(v)$ is even, $(d_{G}(v)+1)/k$ must not be an even integer number,
which implies that $d_{G_{i}}(v)\leq\frac{1}{k}d_{G}(v)+2$. Since
$d_{G^{\prime}}(v)$ is even, if $\frac{1}{k}d_{G}(v)$ is an even integer
number, then $d_{G^{\prime}}(v)\leq\frac{k-1}{k}d_{G}(v)$. Therefore, since
$d_{G_{i}}(v)$ is even, one can conclude that
$d_{G_{i}}(v)<\frac{1}{k}d_{G}(v)+2$ whether $\frac{1}{k}d_{G}(v)$ is an
integer number or not. Similarly, we have
$d_{G_{i}}(v)>\frac{1}{k}d_{G}(v)-2$. Hence the assertion holds.  $\Box$

By replacing Corollary 5.12 in the proof of Theorem 5.3, one can formulate a
parity version of it as the following theorem. It would be interesting to
determine the sharp upper bound on vertex degrees to make another
generalization for Corollary 5.12.

###### Theorem 5.14

. Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ nonnegative real numbers
with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph with even
degrees, then it can be edge-decomposed into $k$ even factors
$G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,

$|d_{G_{i}}(v)-\varepsilon_{i}d_{G}(v)|<6.$

###### Proof.

Apply Corollary 5.12 along with the same arguments stated in the proof of
Theorem 4 in [8].  $\Box$

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Scaling Behavior of Fractional Growth Processes

Reza Torabi Department of Physics and Astronomy, University of Calgary,
Calgary, Alberta T2N 1N4, Canada
e-mail<EMAIL_ADDRESS>

###### Abstract

An exact solution is introduced for one dimensional space-fractional Edwards-
Wilkinson equation. It is shown that the roughness obeys the Family-Viscek
dynamic scaling form and the scaling exponents is derived. It is seen that the
scaling exponents are different from those coming from ordinary Edwards-
Wilkinson equation and depend on the fractional order. Scaling exponents in
the presence of correlated noise is also obtained.

PACS numbers: 89.75.Da; 02.50.Ey; 68.55.-a; 05.40.-a

##  1 Introduction

Fractional calculus [1, 2, 3, 4] deals with the generalization of
differentiation and integration to fractional order. It is a powerful
mathematical tool that is used to describe complex systems in which
conventional approaches have failed. The interest in fractional calculus has
grown rapidly in recent years due to its various applications.

There are many different areas where fractional equations describe real
processes such as anomalous diffusion [5, 6, 7, 8], chaotic dynamics [9, 10],
dynamics in complex media [11, 12, 13, 14, 15], relaxation in complex
viscoelastic materials [16], random processes of Levy type [17, 18, 19], heat
transfer in fractal media [20, 21], acoustic waves in complex media [22], non-
Hamiltonian mechanics [23, 24], theory of long-range interactions [25, 26],
fractional Fokker-Planck equation [27], electromagnetic field of fractal
distribution of charged particles [28], electromagnetic field of fractal media
[29, 30], damped oscillations [31], Navier-Stokes equations on Cantor sets
[32], fractional schrödinger equation [33] and generalized Langevin equation
[34, 35, 36].

In fact, the generalization of many equations of physics to fractional order
has been carried out due to various reasons such as the presence of memory
effects, dissipation, intermittency, fractal media, long-range interactions
and etc. One of these reasons that we are interested in is the presence of
fractal media. Fractional calculus allow us to take into account the fractal
geometry of the medium. In this case the fractal structure is disregarded and
instead, the process is described by a fractional equation [37, 38, 39, 5].
Here we consider the growth process of an interface in a porous (disordered)
medium . Since porous media can be considered as fractal structures, we use
fractional calculus to describe the process.

The growth process of interfaces far from equilibrium, where the height
gradient is negligible, is governed by Edwards-Wilkinson equation. The
dynamics of interface in disordered media is described by adding a quenched
noise to the Edwards-Wilkinson equation. It is hard to study this equation due
to the presence of the quenched noise. Here we present an analytical approach
based on the fractional calculus to find how the exponents change in
disordered media and to see if roughness obeys the dynamic scaling form in
these media.

The paper is organized as follows: In section II, a brief Introduction to the
Edwards-Wilkinson equation both in ordinary and disordered media is presented.
In section III, we introduce a fractional Edwards-Wilkinson equation for
interface in disordered media and solve it analytically. We show that the
roughness of an interface in a disordered medium obeys the Family-Viscek
scaling form and find the scaling exponents. In section IV, the height-height
correlation function is discussed. The fractional Edwards-Wilkinson equation
in the presence of correlated noise is also studied in section V. Finally the
conclusion is presented in section VI.

##  2 Interfaces in disordered media

The dynamics of an interface, where $|\nabla h|\ll 1$, is governed by the
Edwards-Wilkinson equation. This equation is a linear Langevin equation with
the form [40, 41]

$\frac{\partial h({\bf x},t)}{\partial t}=\nu\nabla^{2}h({\bf x},t)+\eta({\bf
x},t),$ (1)

where $h({\bf x},t)$ is the height of a $d$ dimensional interface at point
${\bf x}$ and time $t$. The first term on the right hand side describes
relaxation of the interface and $\nu$ is sometimes called surface tension.
$\eta({\bf x},t)$ is a stochastic noise which reflects the random
fluctuations. It has zero average, $\langle\eta({\bf x},t)\rangle=0$, and its
second momentum is given by

$\langle\eta({\bf x},t)\eta({\bf
x^{\prime}},t^{\prime})\rangle=2D\delta^{d}({\bf x}-{\bf
x^{\prime}})\delta(t-t^{\prime}).$ (2)

Here $\langle\rangle$ denotes average over the noise distribution and Eq. (2)
implies that the noise has no correlation in space and time. The interface
width, $W$, which characterizes the roughness of the interface is defined by
the root mean square fluctuation in the height.

Because of the linear character of the Edwards-Wilkinson equation, one can
solve it straightforwardly and show that the width of the interface obeys the
Family-Viscek dynamic scaling form

$W^{2}(L,t)\simeq L^{2\alpha}F(\frac{t}{L^{z}}),$ (3)

and also obtain the scaling exponents. $\alpha$ and $z$ are known as roughness
and dynamic exponents, respectively. $F(u)$ is a scaling function with the
properties $F(u)\rightarrow const$ as $u\rightarrow\infty$ and
$F(u)\rightarrow u^{(2-d)/2}$ as $u\rightarrow 0$. For $t\ll L^{z}$ the $L$
dependence drops out and (3) yields to $W^{2}(L,t)\simeq t^{2\beta}$ with
$\beta=\alpha/z$. $\beta$ is growth exponent and the scaling exponents of the
Edwards-Wilkinson equation are

$\alpha=\frac{2-d}{2},\;\;\;\beta=\frac{2-d}{4},\;\;\;z=2.$ (4)

The discrete version of the model, governs by (1), is ”random deposition with
surface relaxation”. Surface relaxations allow the deposited particles to
diffuse along the interface up to a finite distance to find the position with
lowest height.

In the case of interface in a disordered medium, a quenched noise, that is
generated by the disorder is added to the Edward-Wilkinson equation as

$\frac{\partial h({\bf x},t)}{\partial t}=\nu\nabla^{2}h({\bf x},t)+\eta({\bf
x},t)+\eta({\bf x},h),$ (5)

It is assumed that $\eta({\bf x},h)$ has zero mean, $\langle\eta({\bf
x},h)\rangle=0$, and the correlation of the form

$\langle\eta({\bf x},h)\eta({\bf
x^{\prime}},h^{\prime})\rangle=\delta^{d}({\bf x}-{\bf
x^{\prime}})\Delta(h-h^{\prime}),$

where $\Delta$ is a function that falls of rapidly when its arguments is
large. Equation (5) can not be solved in a straightforward manner due to the
nature of the quenched noise. To understand the effect of quenched noise,
there has been a lots of numerical studies [40]. In general it is agreed that
quenched noise produce anomalous roughening with a roughness exponent larger
than the value produced by (4).

##  3 The fractional Edward-Wilkinson equation

In disordered media the deposited particle undergoes anomalous diffusion, due
to the effect of medium, to find the position with lowest height. This yields
to an anomalous relaxation process. According to this fact, we introduce a
fractional Edward-Wilkinson equation to describe the process. In other words,
to incorporate the effect of disordered media, we generalize Edward-Wilkinson
equation to fractional order instead of adding the quenched noise. This enable
us to solve the problem via analytical approaches. The generalized Edward-
Wilkinson equation is in the form

$\frac{\partial h({{\bf x}},t)}{\partial
t}=\nu_{\gamma}\frac{\partial^{\gamma}}{\partial|{\bf x}|^{\gamma}}h({{\bf
x}},t)+\eta({{\bf x}},t),$ (6)

where $\gamma$ is the fractional order and $\gamma<2$ due to the porosity.
$\nu_{\gamma}$ is a generalized surface tension and the noise is again a white
one. The fractional derivative is defined via its Fourier transform according
to [17, 34, 42]

$F_{\bf k}\left\\{\frac{\partial^{\gamma}}{\partial|{\bf
x}|^{\gamma}}\right\\}\equiv-|{\bf k}|^{\gamma}.$

Another definition for the above fractional derivative is given in terms of
the Laplacian as ${\partial^{\gamma}}/{\partial|{\bf
x}|^{\gamma}}=-(-\nabla^{2})^{\gamma/2}$ [4]. In fact, this equation is an
special case of the generalized elastic model proposed by [34, 42]. It can be
achieved by considering the hydrodynamic friction kernel to be local and
proportional to the Dirac delta function. As we need, the authors of [34, 42]
showed that, in the systems obeying (6), the particle placed at ${\bf x}$
undergoes a subdiffusive motion (a fractional Brownian motion) due to the
spatial interactions with the other system components.

Equation (6) can be solved in a straight forward manner. Fourier transform of
(6) in space and time gives

$h({\bf k},\omega)=\frac{1}{\nu_{\gamma}k^{\gamma}-i\omega}\eta({\bf
k},\omega),$ (7)

where $k=|{\bf k}|$ and

$h({{\bf k}},\omega)=\int d^{d}{\bf x}\;dt\;e^{-i({\bf k}\cdot{\bf x}-\omega
t)}h({{\bf x}},t),$ $\eta({{\bf k}},\omega)=\int d^{d}{\bf x}\;dt\;e^{-i({\bf
k}\cdot{\bf x}-\omega t)}\eta({{\bf x}},t).$

In order to describe a growth process that is started from a flat substrate at
$t=0$, we set $\eta({\bf x},t)=0$ for $t<0$. The second momentum of the noise
in Fourier space then takes the form

$\langle\eta({\bf k},\omega)\eta({\bf
k^{\prime}},\omega^{\prime})\rangle=2D(2\pi)^{d}\delta^{d}({\bf k}+{\bf
k^{\prime}})\int_{0}^{\infty}d\tau e^{i(\omega+\omega^{\prime})\tau}.$ (8)

The width of an interface of size $L$ is

$W^{2}(L,t)=\left\langle\frac{1}{N}\sum_{\bf x}\left(h({\bf
x},t)-\frac{1}{N}\sum_{\bf x^{\prime}}h({\bf
x^{\prime}},t)\right)^{2}\right\rangle,$

where it can be expressed as [41]

$W^{2}(L,t)=\int\frac{d^{d}{\bf k}}{(2\pi)^{d}}\frac{d^{d}{\bf
k^{\prime}}}{(2\pi)^{d}}\langle h({\bf k},t)h({\bf k^{\prime}},t)\rangle.$ (9)

Thus, in order to find $W^{2}(L,t)$, we need to calculate the quantity
$\langle h({\bf k},t)h({\bf k^{\prime}},t)\rangle$. Using (7), (8) in (9) and
performing the $\omega$ integral, we obtain

$\langle h({\bf k},t)h({\bf
k^{\prime}},t)\rangle=\frac{D}{\nu_{\gamma}k^{\gamma}}(1-e^{-2\nu_{\gamma}k^{\gamma}t}){(2\pi)^{d}}\delta^{d}({\bf
k}+{\bf k^{\prime}}),$ (10)

where we have used the identity

$\int_{-\infty}^{+\infty}\frac{e^{imx}}{n-ix}=-2\pi e^{mn}.$

We note that the relaxation time $\tau$ of modes of wave vector ${\bf k}$ is
now $\tau\simeq{1}/{\nu_{\gamma}k^{\gamma}}$ instead of $\tau\simeq{1}/{\nu
k^{2}}$.

Putting (10) in (9) and carrying out the integral using spherical coordinates
yields

$W^{2}(L,t)=K_{d}(\frac{D}{\nu_{\gamma}})\int_{\frac{2\pi}{L}}^{\frac{\pi}{a}}\frac{dk}{k^{\gamma+1-d}}(1-e^{-2\nu_{\gamma}k^{\gamma}t}),$
(11)

where $K_{d}^{-1}=2^{d-1}\pi^{d/2}\Gamma(d/2)$ and $a$ is the lattice spacing.
Performing the integration by parts, (11) will cast in the Family-Viscek
scaling form

$W^{2}(L,t)=A+\frac{D}{\nu_{\gamma}}L^{\gamma-d}f_{d}(\frac{\nu_{\gamma}t}{L^{\gamma}})+O(e^{-\frac{2\pi\gamma\nu_{\gamma}t}{a^{\gamma}}}),$
(12)

where

$A=\frac{K_{d}}{d-\gamma}(\frac{D}{\nu_{\gamma}})(\frac{\pi}{a})^{d-\gamma},$

and

$f_{d}(x)=\frac{K_{d}}{(\gamma-d)(2\pi)^{\gamma-d}}{\bigg{(}}1-e^{-2(2\pi)^{\gamma}x}+[{2(2\pi)^{\gamma}x}]^{1-\frac{d}{\gamma}}\int_{2(2\pi)^{\gamma}x}^{\infty}y^{\frac{d}{\gamma}-1}e^{-y}dy{\bigg{)}}.$

At large $x$, $f_{d}(x)$ approaches a constant and for $x\ll 1$ we have
$f(x)\propto x^{\frac{\gamma-d}{\gamma}}$,

The scaling exponents can be easily find by comparing (12) with the family-
viscek scaling form (3) as

$\alpha=\frac{\gamma-d}{2},\;\;\;\beta=\frac{\gamma-d}{2\gamma},\;\;\;z=\gamma,$
(13)

where $z=\alpha/\beta$. The fractional order has affected the scaling
exponents through the relation (13).

The fractional order $\gamma=2$ corresponds to the ordinary Edwards-Wilkinson
equation. For fractional order $\gamma>2$ all the scaling exponents take
larger values than the ones coming from the ordinary Edwards-Wilkinson
equation. However, all the exponents have smaller values in the case of
$\gamma<2$.

##  4 Height-height correlation function

Height-height correlation function is another quantity of interest which can
be used to determine scaling exponents. This quantity can be measured
experimentally. It has the form

$C({\bf r},t)\equiv\langle|h({\bf r},t)-({\bf 0},t)|^{2}\rangle.$ (14)

which is written as

$C({\bf r},t)=\int_{\frac{2\pi}{L}}^{\frac{\pi}{a}}\frac{d^{d}{\bf
k}}{(2\pi)^{d}}\frac{2D}{\nu_{\gamma}k^{\gamma}}(1-e^{-2\nu_{\gamma}k^{\gamma}t})[1-cos({\bf
k}\cdot{\bf r})],$ (15)

in momentum space. Using spherical coordinate and performing the angular
integration yields

$C({\bf
r},t)=\frac{2D}{\nu_{\gamma}}K_{d}\int_{0}^{\frac{\pi}{a}}dkk^{d-1-\gamma}(1-e^{-2\nu_{\gamma}k^{\gamma}t})\left[1-\Gamma\left(\frac{d}{2}\right)J_{\frac{d}{2}-1}(kr)\left(\frac{kr}{2}\right)^{1-\frac{d}{2}}\right],$
(16)

for the case of $L\rightarrow\infty$. $r=|{\bf r}|$ and $J_{n}(x)$ is the
Bessel function of order $n$. We are interested in finding the behavior of
height-height correlation function at large $r$. To this end, we use change of
variables $y=kr$ and $x={\nu_{\gamma}t}/{r^{\gamma}}$. The height-height
correlation function takes the form

$C({\bf
r},t)=\frac{2D}{\nu_{\gamma}}r^{\gamma-d}f_{d}(\frac{\nu_{\gamma}t}{r^{\gamma}})+O(1),$
(17)

is the scaling function. Equation (17) states that the height-height
correlation function also has a universal scaling behavior in the presence of
porosity.

##  5 Correlated noise

In the previous sections we assumed that the noise is an uncorrelated one, for
instance, with a gaussian distribution. But it is not the only possible type
of noise in physical systems. In this section we are going to study the growth
of an interface that is governed by a generalized Edwards-Wilkinson equation
in the presence of spatially and temporally correlated noise using the scaling
arguments.

In the case that the noise has long-range spatial and temporal correlations,
we have

$\langle\eta({\bf x},t)\eta({\bf x^{\prime}},t^{\prime})\rangle\sim|{\bf
x}-{\bf x^{\prime}}|^{2\psi-d}|t-t^{\prime}|^{2\phi-1}.$ (18)

where $\psi(\phi)$ characterize the decay of spatial (temporal) correlations.
The case of uncorrelated noise corresponds to $\psi=0$ and $\phi=0$. To see
how the correlated noise influence the growth exponents, we use scaling
arguments.

If the interface is self-affine, then the dynamic equation (6) would not
change by the rescaling

${\bf x}\rightarrow b{\bf x},\;\;\;h\rightarrow b^{\alpha}h,\;\;\;t\rightarrow
t^{z}$ (19)

The noise second momentum rescales as

$\langle\eta(b{\bf x},b^{z}t)\eta(b{\bf
x^{\prime}},b^{z}t^{\prime})\rangle\sim b^{2\psi-d+2z\phi-z}|{\bf x}-{\bf
x^{\prime}}|^{2\psi-d}|t-t^{\prime}|^{2\phi-1},$

which indicates that $\eta\rightarrow
b^{\psi-\frac{d}{2}+z\phi-\frac{z}{2}}\eta$. Thus, after rescaling, the
dynamic equation (6) becomes

$\frac{\partial h({\bf x},t)}{\partial
t}=b^{z-\gamma}\nu_{\gamma}\frac{\partial^{\gamma}}{\partial|{\bf
x}|^{\gamma}}h({{\bf
x}},t)+b^{\psi-\frac{d}{2}+z\phi+\frac{z}{2}-\alpha}\eta({\bf x},t).$ (20)

Since the dynamic equation (6) is invariant under the scaling transformation
(19), the exponents will be

$\alpha=\psi+\gamma\phi+\frac{\gamma-d}{2},\;\;\;\beta=\frac{\psi}{\gamma}+\phi+\frac{\gamma-d}{2\gamma},\;\;\;z=\gamma.$
(21)

As one can see the presence of correlated noise change the scaling exponent.
Note that, in the case of uncorrelated noise we find the exponents of (13).
Equation (21) reflect the fact that the correlated noise increase the
roughness exponent.

##  6 Conclusion

We considered a one dimensional space-fractional Edwards-Wilkinson equation
and solved it exactly. It has been shown that the roughness obeys Family-
Viscek dynamic scaling form in the case of this generalized Edwards-Wilkinson
equation in similar to the ordinary Edwards-Wilkinson equation. We obtained
the scaling exponents and showed that the fractional order affects the these
exponents. Also, the scaling exponents in the presence of correlated noise has
been obtained.

This equation is a linear Langevin equation and is employed in many areas of
physics such as fluid flow in porous media, flux lines in superconductors,
deposition processes, bacterial growth and so on. Therefore, generalization of
ordinary Edwards-Wilkinson equation to a fractional equation can be
interesting due to its application not only in surface growth phenomena but in
many areas of physics.

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