# Searching for $a_{0}(980)$-meson parton distribution function

Zai-Hui Wu  Hai-Bing Fu<EMAIL_ADDRESS>(Corresponding author) Tao Zhong
<EMAIL_ADDRESS>Yu Chen  Ya-Hong Dai  Department of Physics, Guizhou
Minzu University, Guiyang 550025, P.R. China

###### Abstract

In this paper, we calculate the scalar $a_{0}(980)$-meson leading-twist wave
function by using light-cone harmonic oscillator model (LCHO). In which the
model parameters are determined by fitting the $\xi$-moments
$\langle\xi_{a_{0}}^{n}\rangle_{\zeta}$ of its light-cone distribution
amplitudes. Then, the $a_{0}(980)$-meson leading-twist light-cone distribution
amplitudes with three different scales $\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$ are
given. After constructing the relationship between $a_{0}(980)$-meson leading-
twist parton distribution functions/valence quark distribution function and
its LCHO wave function, we exhibit the $\mathpzc q^{a_{0}}(x,\zeta)$ and
$x\mathpzc q^{a_{0}}(x,\zeta)$ with different scales. Furthermore, we also
calculate the Mellin moments of the $a_{0}(980)$-meson’s valence quark
distribution function $\langle x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}$ with
$n=(1,2,3)$, i.e. $\langle x\mathpzc q^{a_{0}}\rangle_{\zeta_{5}}=0.026$,
$\langle x^{2}\mathpzc q^{a_{0}}\rangle_{\zeta_{5}}=0.017$ and $\langle
x^{3}\mathpzc q^{a_{0}}\rangle_{\zeta_{5}}=0.012$. Finally, the scale
evolution for the ratio of the Mellin moments $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ are presented.

###### pacs:

12.38.-t, 12.38.Bx, 14.40.Aq

1\. Introduction — The exploration of quark-gluon structure in hadrons is a
cross-cutting edge issue between particle physics and medium-high energy
nuclear physics in recent years. In which quarks and gluons, called partons,
are the fundamental degrees of freedom of quantum chromodynamics (QCD).
Although the parton can not be directly observed, the QCD factorization
theorem allows one to express the information of the parton inside the nucleon
in terms of nonperturbative functions Collins:1989gx . At the same time, the
parton distribution function (PDF) is considered to be the most important
nonperturbative function, which plays an important role in describing the
nonperturbative QCD for the internal structure of hadronic bound states
Berger:1979du . In addition, it also gives the probability of finding quarks
and gluons inside a hadron. In the infinite momentum coordinate system
Gribov:1973jg ; Bjorken:1969ja ; Feynman:1969ej ; Bjorken:1968dy , the PDFs
are used to describe the one-dimensional momentum distributions of quarks and
gluons. Therefore, the internal structure of hadrons can be studied by
calculating the meson’s PDF.

The PDF constitute the basic limit of Higgs boson representation in terms of
coupling, which is the main system for standard model (SM) measurement such as
$W$-boson mass. And it is also still the largest uncertainty outside the
production of SM heavy particles, so it has important phenomenological value.
The MMHTHarland-Lang:2014zoa , CT Dulat:2015mca , NNPDF NNPDF:2017mvq ,
HERAPDF Alekhin:2017kpj , and JAM Ethier:2017zbq  have made substantial
efforts to determine PDFs and their uncertainties. The pion deemed to the
lightest bound state of QCD and kaon have been predicted by many theoretical
calculations, chiral-quark model Nam:2012vm ; Watanabe:2016lto ;
Watanabe:2017pvl , Nambu-Jona-Lasinio model Hutauruk:2016sug , light-front
holographic QCD (LHFQCD) deTeramond:2018ecg ; Chang:2020kjj ; Watanabe:2019zny
; Lan:2020fno , light front quantization Lan:2019vui ; Lan:2019rba ;
Lan:2020hyb , maximum entropy method Han:2018wsw ; Han:2020vjp , Dyson-
Schwinger equations (DSEs) Chang:2014lva ; Chang:2014gga ; Chang:2021utv ;
Chen:2016sno ; Shi:2018mcb ; Bednar:2018mtf ; Ding:2019lwe ; Freese:2021zne ;
Cui:2021mom ; Cui:2020tdf ; Cui:2022bxn  and lattice QCD Gao:2022iex ;
Gao:2020ito ; Gao:2021dbh ; Sufian:2020vzb ; Lin:2020ssv ; Joo:2019bzr ;
Shugert:2020tgq ; Izubuchi:2019lyk ; Sufian:2019bol ; Zhang:2018nsy  for the
valence quark PDF. In most cases, the light front wave function (LFWF) is
adopted to predict the PDFs. The LFWF is obtained by calculating the Bethe-
Salpeter wave functions by the covariant DSEs and projecting the quantity in
the Bethe-Salpeter wave functions to the light front in Ref. Cui:2022bxn . At
the same time, other models have also analyzed LFWF, such as LHFQCD
deTeramond:2018ecg ; Chang:2020kjj ; Watanabe:2019zny ; Lan:2020fno  and BLFQ
Lan:2019vui ; Lan:2019rba ; Lan:2020hyb . Besides, there are few papers on the
PDFs of the scalar meson $a_{0}(980)$ with the same quark $q\bar{q}$ state,
which motivates us a research motivation.

One of the earliest predictions for $J=0$ meson valence-quark distribution
function for large-$x$ behaviour within the QCD improved parton model
Ezawa:1974wm ; Farrar:1975yb ; Holt:2010vj , has the following expression:

$\displaystyle\mathpzc q^{M}(x;\zeta)\overset{x\simeq 1}{=}\mathpzc
c(\zeta)(1-x)^{\beta_{\zeta}},~{}{\beta_{\zeta}}=2,$ (1)

where $\mathpzc c(\zeta)$ is independent of $x$ and the $\zeta$ stands for the
resolving scale. Since the meson’s PDF can be obtained directly from its wave
function (WF), a more accurate $a_{0}(980)$-meson WF is crucial for us to
determine its PDF. The light-cone harmonic oscillator model (LCHO) for the
light or heavy meson WF is mainly based on the Brodsky-Huang-Lepage (BHL)
prescription BHL , which have been used in many cases Zhong:2014jla ;
Zhong:2021epq ; Wu:2012kw ; Huang:2013gra ; Huang:2013yya ; Zhong:2015nxa ;
Huang:1994dy ; Huang:2004fn ; Wu:2011gf . For this model, the total WF can be
separated into spin-space WF $\chi_{M}^{\lambda_{1}\lambda_{2}}(x,{\bf
k}_{\bot})$ and spatial wave function $\psi^{R}_{M}(x,{\bf k}_{\bot})$. The
spatial WF is divided into $x$-dependence part and ${\bf k}_{\bot}$-dependence
part for calculation. The ${\bf k}_{\bot}$-dependence part derives from the
approximate bound-state solution and the $x$-dependence part $\varphi_{M}(x)$
can be expanded in Gegenbauer polynomials.

Furthermore, the meson light-cone distribution amplitudes (LCDAs) can also
been related to its WF, which leads to the indirect relationship between meson
LCDAs and PDF. In many applications of LCDAs, one usually takes a truncated
form to determine DAs, involving only the first few terms of the Gegenbauer
expansion series. With the increase of $n$, there will exist dimensional
anomalies which lead to spurious oscillations. In addition, one of the most
important factors is the unreliability of the higher-order Gegenbauer moments.
In order to improve this phenomenon, one can adopt LCHO model to deal with the
meson LCDAs. In our previous works Zhong:2021epq ; Hu:2021lkl , meson’s the
leading-twist LCDAs are studied by using the LCHO model, and then the model
parameters are determined by fitting moments with the least squares method.
Therefore, the $a_{0}(980)$-meson PDFs will be studied based on BHL
prescription in this paper.

2\. Theoretical Framework — If one wants to use the typical probability
expression of quantum mechanics to describe the measurable properties of a
given hadron, the first thing one needs to do is finding the WF. Each element
of the WF fock-space decomposition represents the probability amplitude of
finding $n$ components in the hadron. However, the PDFs describe the
longitudinal momentum distribution parton of hadron. To derive the
$a_{0}(980)$-meson leading-twist PDF, the following expression can be used
Cui:2020tdf

$\displaystyle\mathpzc q^{a_{0}}(x,\zeta)=\int_{|{\bf
k}_{\bot}|^{2}\leq\zeta^{2}}d^{2}{\bf k}_{\bot}|\psi_{a_{0}}(x,{\bf
k}_{\bot})|^{2}.$ (2)

where $\psi_{a_{0}}(x,\textbf{k}_{\bot})$ is WF and $\zeta$ stands for the
scale. In order to calculate the PDF, the connection between distribution
amplitudes and distribution function can be established to achieve the
purpose. Exploiting this relationship, one can predict $a_{0}(980)$-meson
leading-twist PDFs with the LCHO model based on BHL description BHL . The LCHO
model of the $a_{0}(980)$-meson leading-twist WF is denoted by

$\displaystyle\psi_{a_{0}}(x,{\bf
k}_{\bot})=\sum_{\lambda_{1}\lambda_{2}}\chi_{a_{0}}^{\lambda_{1}\lambda_{2}}(x,{\bf
k}_{\bot})\psi^{R}_{a_{0}}(x,{\bf k}_{\bot}),$ (3)

with ${\bf k}_{\bot}$ is transverse momentum. What’s more, $\lambda_{1}$ and
$\lambda_{2}$ are the helicities of the two constituent quark. The spin-space
WF $\chi_{a_{0}}^{\lambda_{1}\lambda_{2}}(x,{\bf k}_{\bot})$ comes from the
Wigner-Melosh rotation. The different forms for $\lambda_{1}\lambda_{2}$ can
also be found in Ref. Huang:1994dy . Thus the sum of the spin-space WF have
the following form

$\displaystyle\sum_{\lambda_{1}\lambda_{2}}\chi_{a_{0}}^{\lambda_{1}\lambda_{2}}(x,{\bf
k}_{\bot})=\frac{m_{q}^{2}}{\sqrt{{\bf k}^{2}_{\bot}+m_{q}^{2}}},$ (4)

with $m_{q}=m_{u}=m_{d}$. On the other hand, the BHL description Guo:1991eb ;
Huang:1994dy  proposed the assumption that the valence Fock WF depends only on
the energy variable $\epsilon$ outside the shell. At the same time, the
connection between the equal-time WF in the rest frame and the light-cone wave
function in the infinite frame by equating the energy propagator
$\epsilon=M^{2}-(\sum_{i=1}^{n}k_{i})^{2}$ is proposed. The propagators in
different frames are as follws

$\displaystyle\epsilon_{1}=M^{2}-\left(\sum\limits_{i=1}^{n}q^{0}_{i}\right)^{2},$
$\displaystyle\sum\limits_{i=1}^{n}{\bf q}^{i}=0$
$\displaystyle\epsilon_{2}=M^{2}-\bigg{(}\sum\limits_{i=1}^{n}\frac{{\bf
k}^{2}_{\bot i}+m^{2}_{i}}{x_{i}}\bigg{)},$
$\displaystyle\sum\limits_{i=1}^{n}{\bf k}^{2}_{\bot i}=0,$
$\displaystyle\sum\limits_{i=1}^{n}x^{i}=1.$ (5)

For the two-particle system, one can get

$\displaystyle{\bf q}\leftrightarrow\frac{{\bf k}^{2}_{\bot
i}+m^{2}_{q}}{4x(1-x)}-m^{2}_{q}$ (6)

where $q^{0}_{1}=q^{0}_{2}$. Besides, there is approximate connection between
$\psi_{\rm CM}({\bf q})$ and $\psi_{\rm LC}(x,{\bf k})$,

$\displaystyle\psi_{\rm CM}({\bf q}^{2})\leftrightarrow\psi_{\rm
LC}\bigg{(}\frac{{\bf k}^{2}_{\bot}+m^{2}_{q}}{4x(1-x)}-m^{2}_{q}\bigg{)}$ (7)

Based on the approximation for bound state solution of the meson quark model,
the WF of the harmonic oscillator model in the rest frame is expressed as

$\displaystyle\psi_{\rm CM}({\bf q}^{2})=A\exp\bigg{(}-\frac{{\bf
q}^{2}}{2\beta^{2}}\bigg{)}$ (8)

Combining Eqs. (7) and (8), the spatial WF is

$\displaystyle\psi^{R}_{a_{0}}(x,{\bf
k}_{\bot})=A_{a_{0}}\varphi_{a_{0}}(x)\exp\left[-\frac{{\bf
k}^{2}_{\bot}+m_{q}^{2}}{8\beta_{a_{0}}^{2}x\bar{x}}\right].$ (9)

where $A_{a_{0}}$ is the normalization constant and
$\varphi_{a_{0}}(x)=(x\bar{x})^{\alpha_{a_{0}}}C_{1}^{3/2}(2x-1)$. Thus, one
can get the $a_{0}(980)$-meson leading-twist WF

$\displaystyle\psi_{a_{0}}(x,{\bf
k}_{\bot})=\frac{m_{q}^{2}A_{a_{0}}\varphi_{a_{0}}(x)}{\sqrt{{\bf
k}^{2}_{\bot}+m_{q}^{2}}}\exp\left[-\frac{{\bf
k}^{2}_{\bot}+m_{q}^{2}}{8\beta_{a_{0}}^{2}x\bar{x}}\right].$ (10)

Furthermore, the $a_{0}(980)$-meson leading-twist valence-quark distribution
function can be obtained by integrating over the squared transverse momentum,
i.e. Eq. (2), which leads to the following formula

$\displaystyle\mathpzc
q^{a_{0}}(x,\zeta)=\frac{A^{2}_{a_{0}}m_{q}^{2}(1-2x)^{2}(x\bar{x})^{2\alpha_{a_{0}}}}{512\pi^{5}}$
$\displaystyle\quad\times\left\\{\textrm{Ei}\left[\sqrt{\frac{m_{q}^{2}+\zeta^{2}}{4\beta_{a_{0}}^{2}x\bar{x}}}\right]-\textrm{Ei}\left[\sqrt{\frac{m_{q}^{2}}{4\beta_{a_{0}}^{2}x\bar{x}}}\right]\right\\},$
(11)

with $\textrm{Ei}(x)=-\int^{\infty}_{-z}\frac{e^{-t}}{t}dt$. It guarantees the
relationship

$\displaystyle\int_{0}^{1}dx\mathpzc q^{a_{0}}(x,\zeta)=1$ (12)

which is in agreement with the cases of pion and kaon. For further research,
the process-independent effective charge is used to redesign the process-
dependent-charge alternative and implement evolution to integrate the one-loop
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations with describing
the evolution of quarks or gluons fragmenting into hadrons and identifying
only a hadron at a time. The relevant explanation can be found in Ref.
Cui:2020tdf . Using this process, one can get the Mellin moments of
$a_{0}(980)$-meson’s valence-quark distribution function:

$\displaystyle\langle x^{n}\mathpzc
q^{a_{0}}\rangle_{\zeta}=\int^{1}_{0}x^{n}\mathpzc q^{a_{0}}(x,\zeta)dx$ (13)

Then, the ratio of Mellin moments is also a point of interest, and its
expression is as follows

$\displaystyle\mathpzc x^{n}_{~{}a_{0}}(\zeta,\zeta_{k})=\frac{\langle
x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}}{\langle x^{n}\mathpzc
q^{a_{0}}\rangle_{\zeta_{k}}}$ (14)

Another significant physical quantity associated with $a_{0}(980)$-meson PDF
is its LCDA. The relationship between the $a_{0}(980)$-meson leading-twist
LCDA and the WF is

$\displaystyle\phi_{a_{0}}(x,\zeta)=\int_{|{\bf
k}_{\bot}|^{2}\leq\zeta^{2}}\frac{d^{2}{\bf
k}_{\bot}}{16\pi^{3}}\psi_{a_{0}}(x,{\bf k}_{\bot}).$ (15)

After integrating over the squared transverse momentum, one can get the LCDA
formula

$\displaystyle\phi_{a_{0}}(x,\zeta)=\frac{A_{a_{0}}m_{q}\beta_{a_{0}}}{4\sqrt{2}\pi^{3/2}}\sqrt{x\bar{x}}\varphi_{a_{0}}(x)$
$\displaystyle\quad\times\left\\{\textrm{Erf}\left[\sqrt{\frac{m_{q}^{2}+\zeta^{2}}{8\beta_{a_{0}}^{2}x\bar{x}}}\right]-\textrm{Erf}\left[\sqrt{\frac{m_{q}^{2}}{8\beta_{a_{0}}^{2}x\bar{x}}}\right]\right\\},$
(16)

where ${\rm Erf}(x)=2\int^{x}_{0}e^{-t^{2}}dx/\sqrt{\pi}$ is the error
function. In order to determine the free model parameters $A_{a_{0}}$,
$\beta_{a_{0}}$ and $\alpha_{a_{0}}$, we should use the $\xi$-moments of the
$a_{0}(980)$-meson leading-twist LCDA, which has the following definition

$\displaystyle\langle\xi^{n}_{a_{0}}\rangle_{\zeta}=\int^{1}_{0}\xi^{n}\phi_{a_{0}}(x,\zeta)dx$
(17)

On the other hand, the $\xi$-moments can be calculated by QCD sum rule
approach within background field theory (BFTSR). The two-point correlation
function is taken as

$\displaystyle\Pi_{a_{0}}^{(n,0)}=i\int d^{4}xe^{iq\cdot x}\langle
0|T\\{J_{n}^{V}(x),J_{0}^{S,{\dagger}}(0)\\}|0\rangle,$ (18)

with $n$ will take the odd numbers, while the even order will vanish due to
the $G$-parity. The currents are
$J^{V}_{n}(x)=\bar{q}_{1}(x)/\\!\\!\\!z(iz\cdot\tensor{D})^{n}q_{2}(x)$ and
$J^{S}_{0}(0)=\bar{q}_{1}(0)q_{2}(0)$. The detail calculation process for the
$\xi$-moments are given in our recent paper Wu:2022qqx .

Then, one can adopt the least squares method to fit $\xi$-moments
$\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$ in determining the free model
parameters. The purpose of the least squares method is to obtain the optimal
value of the fitting parameter $\theta$ through minimizing the likelihood
function

$\displaystyle\chi^{2}(\theta)=\sum\limits_{i=1}^{N}\frac{y_{i}-\mu(x_{i},\theta)}{\sigma^{2}_{i}}$
(19)

$\mu(x_{i},\theta)$ is the $a_{0}(980)$-meson $\xi$-moments
$\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$ of combining Eqs. (16) and (17). The
value of $y_{i}$ and its variance $\sigma_{i}$ are defined the value of
$\xi$-moments calculated by QCD sum rule. And beyond that, making use of the
probability density function
$f(y,n_{d})=(1/\Gamma(\frac{n_{d}}{2})2^{n_{d}/2})y^{n_{d}/2-1}e^{-\frac{y}{2}}$
of $\chi^{2}$, one can get the goodness of fit with the following probability
$P_{\chi^{2}}$

$\displaystyle P_{\chi^{2}}=\int^{1}_{0}(f_{y};n_{d})dy$ (20)

with $P_{\chi^{2}}\in(0,1)$. The closer the goodness of fit is to $1$, the
better the parameters are obtained. Incorporating the effect of scale $\zeta$,
according to the renormalization group equations (RGE) of Gegenbauer moments
of the $a_{0}(980)$-meson leading-twist LCDA,

$\displaystyle a_{n}^{a_{0}}(\zeta)$ $\displaystyle=$ $\displaystyle
a_{n}^{a_{0}}(\zeta_{0})E_{n}(\zeta,\zeta_{0}),$ (21)

where
$E_{n}(\zeta,\zeta_{0})=[\alpha_{s}(\zeta)/\alpha_{s}(\zeta_{0})]^{-(\gamma_{n}^{(0)}+4)/b}$
and the coefficient $b=(33-2n_{f})/3$ Cheng:2005nb . $n_{f}$ is the number of
active quark flavors. The $\zeta_{0}$ and $\zeta$ are considered as the
initial scale and the running scale. Here we make a notation that the
$\xi$-moments can translate into $a_{n}$-moment directly. The one-loop
anomalous dimensions is

$\displaystyle\gamma_{n}^{(0)}=C_{F}\bigg{[}1-\frac{2}{(n+1)(n+2)}+4\sum^{n+1}_{j=2}\frac{1}{j}\bigg{]}$
(22)

with $C_{F}=4/3$. Then, one can gain the $\xi$-moments at the arbitrary scales
$\zeta$. Finally, after fitting the moments
$\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$ with the least squares method, we can
predict the $a_{0}(980)$-meson valence quark distribution function, Mellin
moments $\langle x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}$ and ratio $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$.

Table 1: The $a_{0}(980)$-meson leading-twist LCDA moments $\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$ at scale $\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$. | $\zeta_{0}$ | $\zeta_{2}$ | $\zeta_{5}$
---|---|---|---
$\langle\xi^{1}_{a_{0}}\rangle_{\zeta}$ | $-0.307\pm 0.043$ | $-0.212\pm 0.030$ | $-0.158\pm 0.022$
$\langle\xi^{3}_{a_{0}}\rangle_{\zeta}$ | $-0.181\pm 0.034$ | $-0.091\pm 0.017$ | $-0.048\pm 0.009$
$\langle\xi^{5}_{a_{0}}\rangle_{\zeta}$ | $-0.078\pm 0.028$ | $-0.055\pm 0.017$ | $-0.039\pm 0.011$
$\langle\xi^{7}_{a_{0}}\rangle_{\zeta}$ | $-0.049\pm 0.026$ | $-0.035\pm 0.015$ | $-0.025\pm 0.010$
$\langle\xi^{9}_{a_{0}}\rangle_{\zeta}$ | $-0.036\pm 0.024$ | $-0.011\pm 0.012$ | $~{}0.000\pm 0.006$

3\. Numerical analysis — To do the numerical analysis, the following input
parameters are used. The mass of $a_{0}(980)$-meson is taken as
$m_{a_{0}}=0.980\pm 0.020~{}{\rm GeV}$. The current light quark-mass, charm
quark mass, the values of the non-perturbative vacuum condensates, the
continuum threshold $s_{0}$ and the corresponding Borel windows used in
$\xi$-moments BFTSR are consistent with our previous work Wu:2022qqx . The
light quark constitute mass is taken as $m_{q}=250~{}{\rm MeV}$. In this
paper, we take three typical scales, the initial scales $\zeta_{0}=1.0~{}{\rm
GeV}$, the processes scale $\zeta_{2}=2.0~{}{\rm GeV}$ and the scale for
$\pi$-nucleon Drell-Yan experiment Conway:1989fs  or the E615 experiment
$\zeta_{5}=5.2~{}{\rm GeV}$, which agree with the pion cases Cui:2020tdf .

Figure 1: The $a_{0}(980)$-meson leading-twist LCDA $\phi_{a_{0}}(x,\zeta)$ changed with three different scales $\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$. Table 2: The LCHO model parameters $A_{a_{0}}$ (in unit: GeV-1), $\beta_{a_{0}}$ (in unit: GeV), $\alpha_{a_{0}}$ and goodness of fit $P_{\chi^{2}_{\rm min}}$ changed with the factorization scale $\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$. $\zeta$ | $~{}A_{a_{0}}$ | $\beta_{a_{0}}$ | $\alpha_{a_{0}}~{}~{}~{}~{}~{}~{}~{}$ | $P_{\chi^{2}_{\rm min}}$
---|---|---|---|---
$1.0$ | $-260$ | $0.5$ | $-0.43$ | $0.773$
$2.0$ | $-380$ | $0.5$ | $-0.05$ | $0.833$
$5.2$ | $-1580$ | $0.5$ | $~{}~{}0.85$ | $0.129$

Firstly, we list the $a_{0}(980)$-meson leading-twist LCDA $\xi$-moments
$\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$ with three different scales
$\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$ in Table 3. In which, the accurate of our
calculation is up to 9th-order. It can be seen that the absolute value of
$\xi$-moments decreases as the scale $\zeta$ increases. Secondly, the absolute
value of $\xi$-moments decreases with the $n$ increases, which shows that our
calculation has good convergence. Then, we adopt the least squares method to
fit the $\xi$-moments $\langle\xi^{n}_{a_{0}}\rangle_{\zeta}$. Next, the
fitting model parameters with different scale $\zeta$ are given in Table 2.
Based on the experience of other mesons, we take the WF model parameter
$\beta_{a_{0}}=0.5$. Obviously, $A_{a_{0}}$ gradually decreases with the
increment of the scale $\zeta$. However, the goodness of fit $P_{\chi^{2}_{\rm
min}}$ increases firstly and then decreases.

Figure 2: (Color online) The $a_{0}(980)$-meson valence quark distribution
function $\mathpzc q^{a_{0}}(x,\zeta)$ and $x\mathpzc q^{a_{0}}(x,\zeta)$ with
different scales.

With the resultant LCHO model parameters, the curves of $a_{0}(980)$-meson
leading-twist LCDA with three scales $\zeta$ are shown in Fig. 1. The figure
shows that

  * •

The behaivor of the three curves are tend to antisymmetric, which will equal
to zero when the LCDA integrate with respect to $x$, e.g.

$\displaystyle\int_{0}^{1}dx\phi_{a_{0}}(x,\zeta)=0.$ (23)

Meanwhile, the three curves go through the zero at the location $x=0.5$.

  * •

The absolute value of the peaks are decreased with the increase of $\zeta$ and
the $x$-location of the peaks are tend to 0.5 with the $\zeta$ increase. When
the scale tend to infinity i.e. $\zeta\to\infty$, the curve of
$a_{0}(980)$-meson LCDA will tend to asymptotic form
$\phi_{a_{0}}(x,\infty)=0$

Secondly, after taking the LCHO parameters into the $a_{0}(980)$-meson
valence-quark distribution function, e.g. Eq. (11), the predictions of
$\mathpzc q^{a_{0}}(x,\zeta)$ can be obtained. The curves of
$a_{0}(980)$-meson valence-quark distribution function $\mathpzc
q^{a_{0}}(x,\zeta)$ and $x\mathpzc q^{a_{0}}(x,\zeta)$ with different scales
$\zeta$ are shown in Fig. 2, which shows that the value of peaks decrease with
the increase of scale $\zeta$. Since the $a_{0}(980)$-meson leading-twist LCDA
is antisymmetric behavior under $u\to(1-u)$ interchange in the SUf(3) limit,
its valence-quark distribution function $x\mathpzc q^{a_{0}}(x,\zeta)$ is tend
to zero at $x=0.5$. Additionally, valence-quark distribution function tends to
bimodal behavior. In general, the valence-quark distribution functions of pion
and kaon tend to a unimodal behavior Cui:2020tdf .

Table 3: The Mellin moments for $a_{0}(980)$-meson leading-twist distribution function $\langle x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}$ with different scales $\zeta=(1.0,2.0,5.2)~{}\rm{GeV}$. $\zeta$ |  $\langle x\mathpzc q^{a_{0}}\rangle_{\zeta}$ |  $\langle x^{2}\mathpzc q^{a_{0}}\rangle_{\zeta}$ |  $\langle x^{3}\mathpzc q^{a_{0}}\rangle_{\zeta}$
---|---|---|---
$1.0$ |  $0.100$ |  $0.074$ |  $0.062$
$2.0$ |  $0.044$ |  $0.031$ |  $0.024$
$5.2$ |  $0.026$ |  $0.017$ |  $0.012$
Figure 3: (Color online) The predicted ratio of Mellin moments of the
$a_{0}(980)$-meson valence-quark distribution function $\mathpzc
x_{\,a_{0}}^{n}(\zeta,\zeta_{k})$ with three fixed scales
$\zeta_{k}=(1.0,2.0,5.2)~{}{\rm GeV}$ changed with arbitrary scales $\zeta$ in
the range $\zeta=[1,6]~{}{\rm GeV}$. In which, the $n$ is taken as $n=(1,2,3)$
respectively.

Using the meson’s valence quark distribution function, we can get the Mellin
moments $\langle x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}$ of the
$a_{0}(980)$-meson valence-quark distribution function, which are presented in
Table 3. From the table, we can see that the Mellin moments are convergence
with the order $n$ increased. Meanwhile, the Mellin moments are also
convergence with the scale $\zeta$ increased. This agrees with the Mellin
moments of pion’s valence-quark distribution function decreases with the
increase of scale. The greater $\zeta$, the smaller the value of the moments
in Refs. Cui:2020tdf . It proves our prediction of the Mellin moments $\langle
x^{n}\mathpzc q^{a_{0}}\rangle_{\zeta}$ are reasonable.

Finally, we also calculate the ratio $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ of Mellin moments changed with the scale
$\zeta$. The predictions of the ratio of Mellin moments with three fixed
scales $\zeta_{k}=(1.0,2.0,5.2)~{}{\rm GeV}$ and different order $n=(1,2,3)$
are depicted in Fig. 3. From which the $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ are increased with index $n$ before the
point of $\zeta_{k}$, and decreased with $n$ after $\zeta_{k}$. The curves of
$\mathpzc x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ are decreasing with the $\zeta$
increases. The curves will coincide with each other when the scale $\zeta$ and
$\zeta_{k}$ tend to infinity.

Summary — In this paper, we fit moments $\langle x^{n}_{a_{0}}\rangle_{\zeta}$
with the least squares method to obtain the free model parameters $A_{a_{0}}$,
$\beta_{a_{0}}$ and $\alpha_{a_{0}}$ at the scales $\zeta=(1.0,2.0,5.2)~{}{\rm
GeV}$. Meanwhile, the goodness of fit $P_{\chi^{2}_{\rm min}}$ is also given.
Then, we present the curves of $a_{0}(980)$-meson leading-twist LCDA shown in
Fig. 1. After constructing the relationship between $a_{0}(980)$-meson
leading-twist WF and PDFs, the $a_{0}(980)$-meson valence quark distribution
function $\mathpzc q^{a_{0}}(x,\zeta)$ and $x\mathpzc q^{a_{0}}(x,\zeta)$ with
different scales are shown in Fig. 2, which tends to bimodal behavior. The
LCDA and PDFs are tend to zero at the location $x=0.5$ due to the antisymmetry
of the WF. Based on the $a_{0}(980)$-meson valence quark distribution
function, we can get the first three order Mellin moments $\langle x\mathpzc
q^{a_{0}}\rangle_{\zeta}$ of the $a_{0}(980)$-meson valence quark DF shown in
Table 3. Referring to the predicted pion’s Mellin moments, our predicted
result is quite reasonable. At the same time, we also give the ratio $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ of Mellin moments with
$\zeta=(1.0,2.0,5.2)~{}{\rm GeV}$ shown in Fig. 3. The ratio $\mathpzc
x^{n}_{\,a_{0}}(\zeta,\zeta_{k})$ shows a downward tendency with the increase
of $\zeta$.

Acknowledgments — This work was supported in part by the National Natural
Science Foundation of China under Grant No.12265010, No.12265009, the Project
of Guizhou Provincial Department of Science and Technology under Grant
No.ZK[2021]024, the Project of Guizhou Provincial Department of Education
under Grant No.KY[2021]030.

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# Two disjoint cycles in digraphs

Mikołaj Lewandowski  Faculty of Computing and Telecommunications, Poznań
University of Technology, Poznań, Poland
<EMAIL_ADDRESS>,  Joanna Polcyn  Faculty of
Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
<EMAIL_ADDRESS>and  Christian Reiher  Fachbereich Mathematik, Universität
Hamburg, Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.

Bermond and Thomassen conjectured that every digraph with minimum outdegree at
least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was
verified for $k\leq 3$. Here we generalise the question asking for all
outdegree sequences which force $k$ vertex disjoint cycles and give the full
answer for $k\leq 2$.

###### Key words and phrases:

cycles, digraphs, extremal graph theory.

###### 2020 Mathematics Subject Classification:

Primary: 05C35.

##  1\. Introduction

In 1963, Corrádi and Hajnal [CH] proved that every undirected graph with at
least $3k$ vertices and minimum degree at least $2k$ contains $k$ vertex
disjoint cycles. In 1981, Bermond and Thomassen [BT] proposed an analogous
conjecture for digraphs.

######  Conjecture 1.1.

For every positive integer $k$ every digraph with minimum outdegree at least
$2k-1$ contains $k$ vertex disjoint cycles.

They also noted that the complete digraph on $2k-1$ vertices shows that the
bound offered by Conjecture 1.1 is optimal. For $k=1$ the problem is easy and
the case $k=2$ was solved in 1983 by Thomassen [T]. More than two decades
later Lichiardopol, Pór, and Sereni [LPS] managed to solve the case $k=3$ and
for all $k>3$ the problem is wide open. It is known, however, that Conjecture
1.1 holds for tournaments [BLS1, BLS2].

The existence of some finite integer $f(k)$ such that every digraph of minimum
outdegree at least $f(k)$ contains $k$ vertex disjoint cycles was established
by Thomassen [T]. Later Alon [N] proved that it suffices to take $f(k)=64k$.

In this paper we generalise the question and ask for _all_ degree sequences
which force the existence of $k$ vertex disjoint cycles. For instance, our
main result shows that degree sequences of the form ${(1,3,3,3,4,4,...)}$ do
not force $2$ disjoint cycles, but sequences of the form $(1,3,3,3,5,\dots)$
do.

######  Definition 1.2.

For a nonempty and nondecreasing sequence $(d_{1},\ldots,d_{n})$ of
nonnegative integers and a positive integer $k$ the relation

$(d_{1},\ldots,d_{n})\longrightarrow k$

means that every directed graph on $n$ vertices with outdegree sequence
$(d_{1},\ldots,d_{n})$ contains $k$ vertex disjoint cycles. The failure of
this statement is indicated by

$(d_{1},\ldots,d_{n})\mathrel{\mkern
5.5mu\arrownot\mkern-5.5mu}\longrightarrow k\,.$

E.g., Conjecture 1.1 asserts that $d_{1}\geq 2k-1$ implies
$(d_{1},\dots,d_{n})\longrightarrow k$. In analogy with Chvatal’s well-known
graph Hamiltonicity theorem one would hope that given $k$ there is a somewhat
satisfactory characterisation of all outdegree sequences forcing $k$ disjoint
cycles. Since minimum degree conditions are sometimes difficult to maintain in
inductive proofs, one may even speculate whether such a strengthened form of
Conjecture 1.1 could be easier to solve than the original question. For $k=1$
such a characterisation is easily obtained (see Lemma 2.1). The case $k=2$
requires the following concept.

######  Definition 1.3.

Let integers $1\leq r\leq s\leq n$ be given. A sequence $(d_{1},\ldots,d_{n})$
satisfying

  1. ( )

$d_{r}\geq r$, $d_{s}\geq s+1$, and

  2. ( )

if $n\geq 2s-r+2$ and $d_{2s-r+2}=s+1$, then there is an integer
$j\in[2s-r+3,n]$ such that $d_{j}\geq j$

is called _$(r,s)$ -large_. We say that $(d_{1},\ldots,d_{n})$ is _large_ if
it is $(r,s)$-large for some two integers $r\leq s$ in $[n]$.

Our main theorem says that being large and forcing two vertex disjoint cycles
are equivalent properties of outdegree sequences.

######  Theorem 1.4.

Let $(d_{1},\ldots,d_{n})$ be a nonempty, nondecreasing sequence of
nonnegative integers. The relation

$(d_{1},\ldots,d_{n})\longrightarrow 2$

holds if and only if the sequence $(d_{1},\ldots,d_{n})$ is large.

Note that every nondecreasing outdegree sequence $(d_{1},\dots,d_{n})$ with
$d_{1}\geq 3=2\cdot 2-1$ is $(1,1)$-large. So the case $k=2$ of Conjecture 1.1
is a straightforward consequence of Theorem 1.4. It would be very interesting
to find a similar result for three cycles.

### Notation

A _directed graph_ $D$, also called a _digraph_ , is a pair $(V(D),E(D))$,
where $V(D)$ is a set of vertices and $E(D)$ is a set of ordered pairs of
vertices called _arcs_. Digraphs considered here may have loops and 2-cycles,
but no parallel arcs. If an arc $x\rightarrow y$ is present, we say that $x$
_dominates_ $y$, and $x$ _dominates_ a set $W\subseteq V(D)$ if it dominates
each vertex of $W$. The _inneighbourhood_ of $x\in V(D)$ is the set of all
vertices dominating $x$ and the number such vertices is called the _indegree_
of $x$. Similarly, the _outneighbourhood_ and _outdegree_ of $x$ are defined
to be, respectively, the set of all vertices dominated by $x$ and the number
of them.

If $U\subseteq V(D)$ we write $D[U]$ for the subdigraph of $D$ induced by $U$
and $D-U$ for the digraph obtained by deleting $U$ (and all arcs starting or
ending in $U$).

By a cycle we always mean a directed cycle, that is an oriented path starting
and ending at the same vertex. A cycle of length $\ell$ is an _$\ell$ -cycle_.
A 1-cycle is a _loop_. In what follows, the outdegree sequence of every
digraph is nondecreasing.

A _transitive tournament $T_{n}$_ is a digraph whose vertex set can be
enumerated such that $V(T_{n})=\\{v_{1},v_{2},\dots,v_{n}\\}$ and
$E(T_{n})=\\{v_{i}\rightarrow v_{j}\colon{v_{i},v_{j}\in V(T_{n})}\textrm{ and
}i>j\\}.$

##  2\. Preliminary results

We begin by describing all outdegree sequences of vertices of a digraph that
force the existence of a cycle.

######  Lemma 2.1.

The statement $(d_{1},\ldots,d_{n})\longrightarrow 1$ is true if and only if
for some $j\in[n]$ the inequality $d_{j}\geq j$ holds.

###### Proof.

The transitive tournament $T_{n}$ exemplifies $(0,1,\dots,n-1)\mathrel{\mkern
5.5mu\arrownot\mkern-5.5mu}\longrightarrow 1$ and, therefore, the relation
$(d_{1},\ldots,d_{n})\longrightarrow 1$ entails $d_{j}\geq j$ for some
$j\in[n]$.

If, conversely, $d_{j}\geq j$ holds for some $j\in[n]$, we delete the vertices
with outdegrees $d_{1},\ldots,d_{j-1}$. The minimum outdegree of the remaining
digraph is at least $1$ and, hence, it contains a cylce. ∎

Next we show that deleting one term from a sequence cannot destroy its
largeness.

######  Fact 2.2.

Let integers $1\leq r\leq s<n$ be given. If a nondecreasing sequence
${\accentset{\rightharpoonup}{d}=(d_{1},\ldots,d_{n})}$ with $d_{n}<n$ is
$(r,s)$-large, then every sequence
$\accentset{\rightharpoonup}{e}=(e_{1},\ldots,e_{n-1})$ obtained from
$\accentset{\rightharpoonup}{d}$ by deleting one arbitrary term is also
$(r,s)$-large.

###### Proof.

Since $s\leq n-1$ and $e_{i}\geq d_{i}$ for all $i\in[n-1]$, the sequence
$\accentset{\rightharpoonup}{e}$ satisfies ( ) ‣ 1.3. To show that it also
satisfies ( ) ‣ 1.3, assume $n-1\geq 2s-r+2$ and $e_{2s-r+2}=s+1$. This
implies $s+1\leq d_{s}\leq d_{2s-r+2}\leq e_{2s-r+2}=s+1,$ and therefore, in
view of ( ) ‣ 1.3 applied to $\accentset{\rightharpoonup}{d}$, there is an
integer $j\in[2s-r+3,n]$ such that $d_{j}\geq j$. Because of $d_{n}<n$ we
infer $j\leq n-1$, which yields $e_{j}\geq d_{j}\geq j$, as required. ∎

##  3\. Proof of Theorem 1.4

Our goal in this section is to establish Theorem 1.4. Let us first assume
${(d_{1},\ldots,d_{n})\longrightarrow 2}$. The digraph depicted in Figure 3.1
shows $(1,\ldots,n)\not\longrightarrow 2$.

$\mathbf{T}_{n-1}$$v$ Figure 3.1. A transitive tournament $T_{n-1}$ joined by
2-cycles to a single vertex $v$ with a loop. An arc from/to a box goes from/to
every vertex of the box. Each directed cycle in this digraph goes through $v$.

Thus there is a smallest number $s\in[n]$ such that $d_{s}\geq s+1$. Now the
set

$X=\\{i\in[s]\colon d_{i}\geq i\\}$

cannot be empty and there exists $r=\min(X)$. The numbers $r$ and $s$ obey
condition ( ) ‣ 1.3. In case $n\geq 2s-r+2$ the graph shown in Figure 3.2
exemplifies

$({\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}0,1,\ldots,r-2,}{\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}r,r+1,\ldots,s-1,}{\color[rgb]{0.5,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0}\underbrace{s+1,\ldots,s+1}_{s-r+3},}2s-r+2,\ldots,n-1)\not\longrightarrow
2$

and thus they cannot violate ( ) ‣ 1.3.

$\mathbf{T}_{r-1}$$\mathbf{T}_{s-r}$$\mathbf{C}_{s-r+2}$$\mathbf{T}_{n-2s+r-2}$${\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}(0,1,\dots,r-2)}$${\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}(d_{1},d_{2},\dots,d_{r-1})}$${\color[rgb]{0.5,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0}(\mathbf{s+1})}$${\color[rgb]{0.5,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0}(\mathbf{d_{2s-r+2}})}$${\color[rgb]{0.5,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0}(\mathbf{s+1},s+1,\dots,s+1)}$${\color[rgb]{0.5,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0}(\mathbf{d_{s}},\
\
d_{s+1},\dots,d_{2s-r+1})}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}(\mathbf{r},r+1,\dots,s-1)}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}(\mathbf{d_{r}},d_{r+1},\dots,d_{s-1})}$$(2s-r+2,2s-r+3,\dots,n-1)$$(d_{2s-r+3},\
\ d_{2s-r+4},\ \,\dots,\ \ d_{n})$$v$ Figure 3.2. A digraph $D$ obtained from
three transitive tournaments, $T_{r-1}$, $T_{s-r}$, $T_{n-2s+r-2}$, one
directed cycle $C_{s-r+2}$ and one single vertex $v$. An arc from/to a box
goes from/to every vertex of the box. Each directed cycle different from
$C_{s-r+2}$ in $D$ goes through $V(C_{s-r+2})$ and $v$. The outdegree sequence
of vertices of $D$ is described.

To verify the converse direction, we consider a counterexample, given by a
directed graph $D=(V,E)$ on $n$ vertices whose outdegree sequence
$(d_{1},\ldots,d_{n})$ is $(r,s)$-large for two integers $r\leq s$ in $[n]$,
but which fails to contain two disjoint cycles. We may assume that the triple
$(D,r,s)$ has been chosen in such a way that $|V|+|E|+s$ is as small as
possible. The desired contradiction will emerge after nine preliminary claims.

######  Claim 3.1.

We have $r=1$.

###### Proof.

Otherwise delete the vertices with degrees $d_{1},\ldots,d_{r-1}$ from $D$,
thus obtaining a smaller digraph $D^{\star}$. Observe that for every $r\leq
j\leq n$ there are at least $n-(j-1)$ vertices in $D^{\star}$ with outdegree
at least $d_{j}-(r-1)$. Therefore, the nondecreasing outdegree sequence
$(e_{1},\ldots,e_{n-r+1})$ of $D^{\star}$ has the property $e_{k}\geq
d_{k+r-1}-(r-1)$ holds for every $k\in[n-r+1]$. Consequently,
$(D^{\star},1,s-r+1)$ is another counterexample to our claim that contradicts
the minimality of $(D,r,s)$. ∎

Notice that now our conditions ( ) ‣ 1.3 and ( ) ‣ 1.3 read

  1. ( )

$d_{1}\geq 1$, $d_{s}\geq s+1$, and

  2. ( )

if $n\geq 2s+1$ and $d_{2s+1}=s+1$, then there is an integer $j\in[2s+2,n]$
such that $d_{j}\geq j$.

Exploiting the minimality of $|E|$ we infer

  1. ( )

If $i\in[s-1]$, then $d_{i}=1$.

  2. ( )

If $i\in[s,\min(2s,n)]$, then $d_{i}=s+1$.

######  Claim 3.2.

There are no loops in $D$. In particular, $d_{n}<n$.

###### Proof.

Assume that $D$ possesses a loop at some vertex $x$. Then by ( ) ‣ 3 the
directed graph $D^{\star}=D-x$ has at most $s-1$ vertices whose outdegree is
smaller than $s$ and, moreover, as $n\geq d_{n}\geq s+1$, it has a further
vertex whose outdegree is at least $s$. So by Lemma 2.1 $D^{\star}$ contains a
cycle, which together with the loop at $x$ yields two disjoint cycles in $D$.
∎

Combining the above claim with ( ) ‣ 3 one gets $s+1\leq d_{s}\leq d_{n}<n$,
and thereby

$n\geq s+2.$ (3.1)

######  Claim 3.3.

Every $2$-cycle of $D$ is dominated by a vertex of outdegree $s+1$.

###### Proof.

Note that (3.1) together with ( ) ‣ 3 tell us that $D$ has at least three
vertices whose outdegrees are at least $s+1$. If $x\longleftrightarrow y$ is a
$2$-cycle in $D$ not dominated by any vertex of outdegree $s+1$, then for the
same reason as before the digraph $D-\\{x,y\\}$ has to contain a cycle. ∎

######  Claim 3.4.

Suppose that an arc $x\rightarrow y$ of $D$ does not appear in a $2$-cycle.

  1. (0 )

There is some vertex $a\not\in\\{x,y\\}$ dominating $x$ and $y$.

  2. (0 )

If the outdegree of $x$ is $1$, then at least $s+1$ vertices distinct from $y$
and having outdegree $s+1$ dominate $\\{x,y\\}$.

###### Proof.

We construct a digraph $D^{\star}$ from $D-x$ by adding all arcs of the form
$z\rightarrow y$, $z\in V(D^{\star})\smallsetminus\\{y\\}$, whenever
$z\rightarrow y\notin E(D)$ and $z\rightarrow x\in E(D)$ (see Figure 3.3).
Plainly $D^{\star}$ cannot contain two disjoint cycles. Observe that the only
vertices whose outdegree is smaller in $D^{\star}$ than in $D$ are those
dominating $\\{x,y\\}$. Their outdegree dropped by one.

$x$$y$$z$$D$$D^{\star}$$x$$y$$z$$\Rightarrow$ Figure 3.3. New arcs in
$D^{\star}$.

If (0 ) fails, then the outdegree sequence of $D^{\star}$ is the same as that
of $D$ with the term corresponding to $x$ removed and thus, in view of Fact
2.2 and Claim 3.2, it is $(r,s)$-large, meaning that $(D^{\star},r,s)$
contradicts the supposed minimality of $(D,r,s)$.

Let us now suppose that $x$ has outdegree $1$. Due to ( ) ‣ 3 this is only
possible if $s>1$. By ( ) ‣ 3 the digraph $D^{\star}$ has $s-2$ vertices whose
outdegree is $1$, while ( ) ‣ 3 shows that each other vertex has the outdegree
at least $s$. Now the only possibility for the triple $(D^{\star},1,s-1)$ not
to contradict the supposed minimality of $(D,1,s)$ is that clause ( ) ‣ 3
fails, which means, in particular, that $n-1\geq 2(s-1)+1$, i.e., $n\geq 2s$,
and that $d_{2s-1}=s$ meaning $D^{\star}$ has $(2s-1)-(s-2)=s+1$ vertices of
outdegree $s$. In $D$ these vertices need to have outdegree $s+1$ and thus
they dominate $\\{x,y\\}$. This proves (0 ). ∎

######  Claim 3.5.

The inneighbourhood of every vertex of $D$ contains a cycle.

###### Proof.

Let $y$ denote any vertex of $D$ and let $N$ be its inneighbourhood.

Assume first that $N=\varnothing$. Then the outdegree sequence of
$D^{\star}=D-y$ is obtained from that of $D$ by removing the term
corresponding to $y$ and, therefore, in view of Fact 2.2 and Claim 3.2, is
$(1,s)$-large. Thus, the triple $(D^{\star},1,s)$ contradicts the minimality
of $(D,1,s)$.

Thereby we have shown $N\neq\varnothing$. By Lemma 2.1 our claim can only fail
if some vertex $x$ of $D[N]$ has no inneighbours in that graph. By Claim 3.3
this is only possible if there is no arc from $y$ to $x$, but this contradicts
Claim 3.4(0 ). ∎

######  Claim 3.6.

There is a $2$-cycle in $D$.

###### Proof.

Otherwise, in view of Claim 3.5, each vertex of $D$ has indegree at least $3$.
Let $D^{\star}=(V,E^{\star})$ be a directed graph arising from $D$ by first
reversing all arrows and then deleting an arbitrary arc. Its outdegree
sequence $(e_{1},\ldots,e_{n})$ has the properties $e_{1}\geq 2$ and
$e_{3}\geq e_{2}\geq 3$ and thus it is $(1,1)$-large. Since
$|V|+|E^{\star}|+1<|V|+|E|+s$, the minimality of $(D,r,s)$ tells us that
$D^{\star}$ and thus also $D$ contains two disjoint cycles, which is a
contradiction. ∎

######  Claim 3.7.

In $D$ there are a directed cycle $C$ all of whose vertices have outdegree
$s+1$ and a vertex $x\not\in V(C)$ connected to every vertex of $C$ by a
$2$-cycle.

###### Proof.

Let us first assume that $D$ has a vertex $x$ belonging to all $2$-cycles.
Define

$\displaystyle A$ $\displaystyle=\\{z\in V\colon z\neq x\text{ and
}x\longleftrightarrow z\text{ is a $2$-cycle}\\}$ $\displaystyle\text{ and }B$
$\displaystyle=\\{z\in A\colon\text{the outdegree of $z$ is $s+1$}\\}.$

Claim 3.6 informs us that $A\neq\varnothing$. Consider any $a\in A$. By Claim
3.3 the $2$-cycle $a\longleftrightarrow x$ is dominated by some vertex
$b\not\in\\{a,x\\}$ whose outdegree is $s+1$. The inneighbourhood of $b$
contains some cycle by Claim 3.5. This cycle cannot be disjoint to
$a\longleftrightarrow x$ and thus there is an arc from $a$ or $x$ to $b$. The
former is impossible, for then the $2$-cycle $a\longleftrightarrow b$ would
not pass through $x$ and for this reason we must have $b\in B$ (see Figure
3.4).

$x$$a\quad a\in A$$b\quad d^{+}_{D}(b)=s+1$ Figure 3.4. A 2-cycle
$a\longleftrightarrow x$ and a vertex $b$ dominating $\\{a,x\\}$.

We have thus shown that every member of $A$ has an inneighbour in $B\subseteq
A$. In particular, $B\neq\varnothing$ and in the restriction $D[B]$ every
vertex has indegree at least $1$. So, in view of Lemma 2.1, there is some
cycle $C$ in $B$ and the claim follows in this case.

From now on we may assume that $D$ does not have a vertex belonging to all of
its $2$-cycles. This means that the undirected graph $G=(V,E_{G})$ the edges
of which correspond to the $2$-cycles of $D$ is not the disjoint union of a
(possibly degenerate) star and isolated vertices. As $G$ cannot have two
independent edges either, it has to be the disjoint union of a triangle and
isolated vertices. In other words, there are three distinct vertices $x$, $y$,
and $z$ of $D$ such that $x\longleftrightarrow y$, $x\longleftrightarrow z$,
and $y\longleftrightarrow z$ are $2$-cycles.

If both $y$ and $z$ have outdegree $s+1$ the claim still holds with
$y\longleftrightarrow z$ playing the rôle of $C$. So let us finally assume
that the outdegree of $y$ is not $s+1$. Applying Claim 3.3 to
$x\longleftrightarrow z$ we get a vertex $a\not\in\\{x,y,z\\}$ dominating $x$
and $z$. Since the cycle contained in the inneighbourhood of $a$ has to
intersect $\\{x,z\\}$ we may further suppose that $x$ dominates $a$. But now
$a\longleftrightarrow x$ and $y\longleftrightarrow z$ are two disjoint
$2$-cycles in $D$ (see Figure 3.5).

$x$$y$$z$$a$ Figure 3.5. 2-cycles $x\longleftrightarrow y$,
$y\longleftrightarrow z$, $y\longleftrightarrow x$ in $D$ and a vertex $a$
dominating $\\{x,z\\}$.

∎

From now on, we fix $C$ and $x$ as obtained by the previous claim. Observe
that the cycle $C$ has to be induced, for otherwise $D$ would contain two
disjoint cycles.

######  Claim 3.8.

We have $s>1$.

###### Proof.

Assume $s=1$. If $D$ has no vertices besides $x$ and those on $C$, then the
outdegree sequence of $D$ has to be

$(\underbrace{2,\ldots,2}_{n-1},n-1)$

for some $n\geq 3$, contrary to ( ) ‣ 3. So there is a vertex $y\neq x$ not
lying on $C$. By Claim 3.5 the inneighbourhood of $y$ contains a cycle. This
cycle needs to have a common vertex $z$ with $C$. But now $x$, $y$ and the
successor of $z$ on $C$ are three distinct outneighbours of $z$, contrary to
the fact that the members of $C$ have outdegree $s+1=2$ (see Figure 3.6).

$x$$z$$y$$C$ Figure 3.6. The cycle $C$ and the vertices $x$ and $y$.

∎

######  Claim 3.9.

If $n\geq 2s+1$, then $d_{2s+1}\geq s+2$. In particular, $D$ has at most $s+1$
vertices of outdegree $s+1$.

###### Proof.

Assume for the sake of contradiction that $n\geq 2s+1$ and $d_{2s+1}<s+2$.
Since $d_{2s+1}\geq d_{s}=s+1$, we have $d_{2s+1}=s+1$, meaning that ( ) ‣ 3
yields the existence of some $j\in[2s+2,n]$ with $d_{j}\geq j$. Let
$D^{\star}$ be the digraph that arises from $D$ if its $s-1$ vertices of
outdegree $1$ are removed. The nondecreasing outdegree sequence
$(e_{1},\ldots,e_{n-s+1})$ of $D^{\star}$ satisfies $e_{1}\geq 2$ and
$e_{j-s+1}\geq j-s+1$. Since $j-s+1\geq s+3\geq 4$ this tells us that the
triple $(D^{\star},1,1)$ violates the minimality of $(D,1,s)$. ∎

Notice that the graph $D-V(C)$ cannot contain a cycle. Since it is not empty,
it needs to contain some vertex $z$ without any outneighbours by Lemma 2.1.

If $z$ had outdegree $1$ in $D$, then $z\neq x$ and its unique outneighbour
$a$ had to lie on $C$. In particular, the outdegree of $a$ had to be $s+1$,
which in view of Claim 3.4(0 ) entailed that $D$ had at least $s+2$ vertices
the outdegree of which is $s+1$, contrary to Claim 3.9.

By ( ) ‣ 3, ( ) ‣ 3, this tells us that the outdegree $\ell$ of $z$ in $D$ is
consequently at least $s+1$. Since $z$ has no outneighbours in $D-V(C)$, the
length of $C$ needs to be at least $\ell$ and possibly together with $z$
itself this yields $s+2$ vertices of $D$ whose outdegree is $s+1$. We have
thereby reached the final contradiction that concludes the proof of Theorem
1.4. ∎

## References