# Mapping Observations of complex organic molecules around Sagittarius B2 with
ARO 12m telescope

Juan Li,1,2 Junzhi Wang,1,2 Haihua Qiao,3,1,2 Donghui Quan,4,5 Min Fang,6
Fujun Du,2,7 Fei Li,1,2,8 Zhiqiang Shen,1,2 Shanghuo Li,1,2,8 Di Li,9,10 Yong
Shi,11,12,13 Zhiyu Zhang,11,12,13 and Jiangshui Zhang,14

1Department of Radio Science and Technology, Shanghai Astronomical
observatory, 80 Nandan RD, Shanghai 200030, China
2Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, China
3National Time Service Center, Chinese Academy of Sciences, Xi’An, Shaanxi,
710600, PR China
4Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science
1-Street, Urumqi 830011, PR China
5Department of Chemistry, Eastern Kentucky University, Richmond, KY 40475, USA
6Department of Astronomy, University of Arizona, 933 North Cherry Avenue,
Tucson, AZ 85721, USA
7Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210034, PR
China
8University of Chinese Academy of Sciences, 19A Yuquanlu, Beijing 100049, PR
China
9 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Road, Chaoyang District, Beijing 100012, PR China
10 CAS Key Laboratory of FAST, Beijing, CN 11School of Astronomy and Space
Science, Nanjing University, Nanjing 210093, China
12Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China
13Collaborative Innovation Center of Modern Astronomy and Space Exploration,
Nanjing 210093, China
14Center for Astrophysics, Guangzhou University, Guangzhou, 510006, PR China
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We performed high-sensitivity mapping observations of several complex organic
molecules around Sagittarius B2 with ARO 12m telescope at 3-mm wavelength.
Based on their spatial distribution, molecules can be classified as either
“extended" that detected not only in Sgr B2(N) and Sgr B2(M), or “compact"
that only detected toward or near to Sgr B2(N) and Sgr B2(M). The “extended"
molecules including glycolaldehyde (CH2OHCHO), methyl formate (CH3OCHO),
formic acid (t-HCOOH), ethanol (C2H5OH) and methyl amine (CH3NH2), while the
“compact" molecules including dimethyl ether (CH3OCH3), ethyl cyanide
(C2H5CN), and amino acetonitrile (H2NCH2CN). These “compact" molecules are
likely produced under strong UV radiation, while “extended" molecules are
likely formed under low-temperature, via gas-phase or grain surface reactions.
The spatial distribution of “warm" CH2OHCHO at 89 GHz differ from the spatial
distribution of “cold" CH2OHCHO observed at 13 GHz. We found evidence for an
overabundance of CH2OHCHO compared to that expected from the gas-phase model,
which indicates that grain-surface reactions are necessary to explain the
origin of CH2OHCHO in Sagittarius B2. Grain-surface reactions are also needed
to explain the correlation between the abundances of “cold" CH2OHCHO and
C2H5OH. These results demonstrate the importance of grain-surface chemistry in
the production of complex organic molecules.

###### keywords:

ISM: abundances - ISM: clouds - ISM: individual (Saggitarius B2) - ISM:
molecules - radio lines: ISM

††pubyear: 2019††pagerange: Mapping Observations of complex organic molecules
around Sagittarius B2 with ARO 12m telescope–References

##  1 Introduction

Complex organic molecules (COMs) are regarded to be related to the origin of
life on Earth (Herbst & van Dishoeck, 2009). Thus, understanding property and
formation mechanism of prebiotic molecules is key for study of astrobiology.
It has been found that prebiotic molecules are very abundant in the Galactic
center giant molecular cloud Sagittarius B2 (Sgr B2) (Menten, 2011). Sgr B2
contains two main sites of star formation, Sgr B2(N) and Sgr B2(M), which have
been the best hunting ground for prebiotic molecules in the interstellar
medium (ISM) since the early 1970s (Belloche et al., 2013). Many COMs are
first detected toward Sgr B2(N), such as glycoaldehyde CH2OHCHO (Hollis et
al., 2000), ethylene glycol HOCH2CH2OH (Hollis et al., 2002), amino
acetonitrile H2NCH2CN (Belloche et al., 2008) and so on. Some COMs, such as
glycolaldehyde and ethylene glycol, were found to be widespread in the
Galactic center (Hollis et al., 2001; Requena-Torres et al., 2006, 2008; Li et
al., 2017).

Theoretical and experimental studies (Garrod et al., 2008; Woods et al., 2013;
Skouteris et al., 2018; Öberg et al., 2009; Fedoseev et al., 2015; Chuang et
al., 2016; Meinert et al., 2016) show that COMs may be formed on cosmic dusts
by radicals that are produced on dust grain surfaces by UV or cosmic ray
induced photons. The COMs are then desorbed into the gas phase due to thermal
and/or non-thermal processes (Garrod & Herbst, 2006). However, these complex
ice compositions cannot be directly observed with telescopes in the infrared;
thus, most constraints come from millimeter observations of desorbed ice
chemistry products. On the other hand, recent quantum chemical computations
show that the combination of radicals trapped in amorphous water ice does not
necessarily lead to the formation of COMs (Enrique-Romero et al., 2016).
Recently, a new scheme for the gas-phase synthesis of glycolaldehyde, a
species with a prebiotic potential, has been proposed. The predicted abundance
of glycolaldehyde agrees well with that measured in solar-type hot corinos and
shock sites (Skouteris et al., 2018). Thus, how these species are synthesized
is a mystery.

Previous mapping observations mainly concentrate on Sgr B2(N) with
interferometers like ATCA and ALMA at small spatial scales ($\leq$ 1 ′) (Corby
et al., 2015; Sánchez-Monge et al., 2017; Belloche et al., 2016, 2019; Bonfand
et al., 2019; Xue et al., 2019), and only a handful of large-scale spectral
line mapping observations toward Sgr B2 has been performed (Jones et al.,
2008, 2011; Chengalur & Kanekar, 2003). The large-scale spatial distribution
of COMs like C2H5CN, HCOOH are still unclear mainly because of their weak
emission (Halfen, Woolf & Ziurys, 2017). Single dish telescopes have ability
to probe the structure of weak lines from extended gas of Sgr B2 complex on
spatial scales much larger than 1′. In this paper we present high-sensitivity
grid imaging results of several COMs around Sgr B2 with the Arizona Radio
Observatory (ARO) 12 m telescope. In §2, we describe the observations and data
reduction. In §3, we present the mapping result, and spatial distribution of
column densities, rotational temperatures, and estimated abundances for the
observed COMs. In §4, we discuss molecular formation pathways and compare our
results with protostars and then summarize our conclusions in §5.

##  2 Observations and data reduction

We performed point by point mapping observations of COMs at 3-mm wavelength
around Sgr B2 in 2017 April and May with the Arizona Radio Observatory 12 m
telescope at Kitt Peak, Arizona. The 85 - 116 GHz Sideband Separating ALMA
Band-3 Receiver was used. The beam size is 70 arcsec at 90 GHz. The 2 IF modes
of Millemeter Auto Correlator (MAC) were employed as backends. The MAC has
4096 channels with 195 kHz sampling interval, corresponding to 0.65 km s-1at
90 GHz. Though the velocity resolution is not high enough for study of
molecular emission from hot cores, but enough for single dish observations of
Sgr B2. The FWHM linewidth is up to $\sim$ 20 km s-1 (Rivilla et al., 2018,
2019) due to the extended envelope and several unresolved cores with different
velocities and linewidths within the beam. The temperature scale was
determined by the chopper wheel method, and is related to the main beam
brightness temperature with main beam efficiency of 0.88. The system
temperature range from 90 to 150 K. The integration time for each position
range from 50 minutes to 90 minutes.

Observations were conducted in position-switching mode, and an offset of +30
arcmin in azimuth was used. The data processing was conducted using GILDAS
software package111http://www.iram.fr/IRAMFR/GILDAS., including CLASS and
GREG. Linear baseline subtractions were used for most of the spectra. For each
transition, the spectra of subscans, including two polarizations, were
averaged to reduce rms noise levels. Gaussian fitting is used to derive the
physical properties of molecule lines, including peak intensity, VLSR, FWHM
line width, and integrated intensity.

##  3 Observing Results

We detected emission lines from t-HCOOH, C2H5CN, CH3OCH3, CH3OCHO, H2NCH2N,
CH2OHCHO, CH3NH2, and C2H5OH toward many positions at a rms level of $\sim$ 3
mK. Figure 1 presents spectra observed toward several positions with ARO 12m
telescope. As is shown in Figure 1, some molecule lines are only detected
toward Sgr B2(N) and Sgr B2(M). Table 1 presents detected transitions, and
their spectroscopic properties in Sgr B2(N). The molecular spectroscopic
parameters are taken from the current public databases: the CDMS
catalog222https://www.astro.uni-koeln.de/cdms/catalog (Müller et al., 2005),
the JPL catalog333https://spec.jpl.nasa.gov (Pickett et al., 1998) and the
Spectral Line Atlas of Interstellar Molecules (SLAIM) database, which are
available in the SPLATALOGUE spectroscopy
database444https://www.splatalogue.net.. Note that the strong C2H5CN emission
near to t-HCOOH 4(0,4)-3(0,3) (89.57917 GHz) make it difficult to determine
the baseline and intensity of t-HCOOH. Only t-HCOOH emission toward Sgr B2(N)
was affected since the C2H5CN emission was very weak toward other positions.

Table 1: Identified Transitions and their observed parameters in Sgr B2(N) Species | Transitions | Rest Freq. | $E_{u}$ |  $\mu^{2}$S | $T_{mb}$ |  VLSR | $\Delta V$ |  $\int T_{mb}$ dv
---|---|---|---|---|---|---|---|---
|  | (GHz) | (K) |  (D2) | (mK) |  (km s-1) |  (km s-1) |  (mK km s-1)
t-HCOOH∗ | 4(0,4)-3(0,3) | 89.57917(1e-6) | 10.76 | 8.077 |  $\sim$100 |  $\sim$64 |  $\sim$20 |  $\sim$2000
C2H5CN,v=0 | 10(9,1)-9(9,0) | 89.584987(5e-5) | 109.32 | 28.16 | blended | - | - | -
C2H5CN,v=0 | 10(9,2)-9(9,1) | 89.584987(5e-5) | 109.32 | 28.16 | blended | - | - | -
C2H5CN,v=0∗ | 10(4,7)-9(4,6) | 89.590028(3e-6) | 41.44 | 124.52 | 160(45) | 63.0(.6) | 22.0(1.4) | 3520(610)
C2H5CN,v=0 | 10(4,6)-9(4,5) | 89.591013(3e-6) | 41.44 | 124.52 | 160(45) | 63.0(.6) | 22.0(1.4) | 3520(610)
C2H5CN,v=0 | 10(3,8)-9(3,7) | 89.628485(3e-6) | 33.66 | 134.89 | 200(10) | 65.0(1.4) | 21.9(4.0) | 4666(654)
C2H5CN,v20=1 | 10(8,3)-9(8,2) | 89.643412(2e-6) | 630.76 | 52.43 | blended |  |  |
C2H5CN,v20=1 | 10(4,7)-9(4,6) | 89.661002(5e-5) | 577.94 | 122.32 | 30(3) | 67.2(.4) | 7.0(1.2) | 211(57)
C2H5CN,v=0 | 10(3,7)-9(3,6) | 89.684710(3e-6) | 33.66 | 134.88 | 134(3) | 60.9(2.6) | 26.8(2.6) | 3821(236)
CH3OCH3,v=0 | 2(2,1)-2(1,2)AE | 89.697737(1e-6) | 8.36 | 8.475 | 91(8) | 66.5(2.6) | 18.4(2.6) | 1783(158)
CH3OCH3,v=0∗ | 2(2,1)-2(1,2)EE | 89.699809(1e-6) | 8.36 | 22.31 | 50(8) | 64.2(2.6) | 15.1(2.6) | 797(158)
CH3OCH3,v=0 | 2(2,1)-2(1,2)AA | 89.702829(2e-6) | 8.36 | 14.131 | 46(8) | 65.2(.3) | 23.0(2.6) | 1131(158)
CH3OCHO, vt=1 | 8(0, 8) - 7(0, 7) A | 89.731695(1e-5) | 207.83 | 41.664 | 29(6) | 64.2(.3) | 7.4(.7) | 228(19)
C2H5CN,v20=1 | 10(3,7) - 9(3,6) | 89.754402(5e-5) | 570.24 | 132.51 | 43(5) | 65.9(.4) | 34.3(1.2) | 1579(46)
H2NCH2CN∗ | 10(0,10)-9(3,6) | 89.770285(7e-6) | 19.50 | 66.296 | 25(3) | 63.7(0.7) | 14.5(1.6) | 377(40)
t-HCOOH | 4(2,3)-3(2,2) | 89.861484(2e-6) | 23.51 | 5.93 | 50(3) | 66.9(.2) | 18.5(.6) | 987(26)
CH2OHCHO∗ | 9(1,8)-8(2,7) | 89.868639(6e-6) | 26.43 | 20.565 | 17(3) | 61.5(2.6) | 18.6(2.6) | 338(41)
t-HCOOH | 4(3,2)-3(3,1) | 89.94821(2e-6) | 35.09 | 3.536 | blended |  |  |
CH3NH2∗ | 1(1)A1-1(0)A2 | 89.956068(2e-5) | 6.45 |  1.5a | 31(3) | 67.8(1.3) | 30.3(2.8) | 982(87)
CH3CH2C13N | 10(2,8)-9(2,7) | 89.992750(1.2e-6) | 28.10 | 139.763 | blended |  |  |
g-C2H5OH | 13(1,12)-13(1,13),g+ | 90.0077(5e-5) | 135.11 | 13.99 | 28(3) | 64.0(.5) | 6.2(1.2) | 188(53)
13CH2CHCN | 10(1,10)-9(1,9) | 90.032588(1.2e-6) | 25.92 | 432.232 | blended |  |  |
CH${}_{3}^{13}$CH2CN, v=0 | 10(2,8)-9(2,7) | 90.079288(1.3e-6) | 28.00 | 139.77 | 24(3) | 66.3(6) | 17.7(1.3) | 457(29)
C2H5OH∗ | 4(1,4)-3(0,3) | 90.11761(2.8e-6) | 9.36 | 5.21 | 266(9) | 68.5(.1) | 22.4(.2) | 6330(37)
CH3OCHO∗ | 7(2,5)-6(2,4) E | 90.145634(2.4e-5) | 19.68 | 34.275 | 86(7) | 65.7(.1) | 16.7(.4) | 1523(28)
CH3OCHO | 7(2,5)-6(2,4) A | 90.156473(1e-5) | 19.67 | 34.289 | 80(7) | 66.3(.4) | 13.7(.9) | 1166(65)

Notes.- ∗: Lines used for mapping. Col. (1): chemical formula; Col. (2):
transition quantum numbers, Col. (3): rest frequency; Col. (4): upper state
energy level (K); Col. (5): dipole-weighted transition dipole matrix elements;
Col. (6): peak temperature; Col. (7): centroid velocity; Col. (8): FWHM
linewidth; Col. (9): integrated intensity. a: $\mu^{2}$S is taken from
(Motiyenko et al., 2014) for CH3NH2.

Figure 1: Spectra obtained toward Sgr B2(N), Sgr B2(M) and (540, -300) away
from Sgr B2(N) with the ARO 12m telescope in the main-beam temperature scale.
The line names are labeled. The “extended" molecular line profiles were
colored with red, while the “compact" molecular line profiles were colored
with blue in spectra of Sgr B2N.

The many hazards of searching for new molecules have been pointed out by
Snyder et al. (2005). These authors proposed some of the criteria needed for
secure molecular identification, including accurate rest frequencies and
consistency among the acquired data set. Halfen et al. (2006) extended
Criteria in Snyder et al. (2005) and applied them to the ARO 12m observations.
They suggest that for an accurate identification, there must be strong
evidence for emission at all favorable, physically connected transitions over
a sufficiently large wavelength range. There cannot be ‘missing’ favorable
lines. Based on these criteria, CH2OHCHO 9(1,8)-8(2,7) (89.868639 GHz), the
glycolaldehyde transition used in this work was found to be strong and
relatively clean. Xue et al. (2019) proposed a quantitative method to identify
weak and mostly uncontaminated transitions. They used P factor, which is the
product of the observed and simulated line profiles, to characterizing the
frequency agreement. They used D factor, which is the ratio between the
difference of the integrated intensities of the compared spectra and the
maximum of the two subtracted from 1, to measure the difference of the line
intensities. For CH2OHCHO 9(1,8)-8(2,7) at 89.868639 GHz, the P factor is 94%,
while the D factor is 16.8%. For CH3OCHO 7(2,5)-6(2,4) E at 90.145 GHz, the P
factor is 99.8%, while the D factor is 25.4% (see appendix in Xue et al.
(2019)). Based on the P factors, both of these two transitions are assigned to
be unblended transition candidates. The low D factors are likely to be caused
by the high excitation temperature (190 K) adopted by Xue et al. (2019).
Transitions of other molecules have been identified by Belloche et al. (2013).
Velocity integrated images of strong and relatively clean transitions,
including CH2OHCHO 9(1,8)-8(2,7), CH3OCHO 7(2,5)-6(2,4)E, t-HCOOH
4(0,4)-3(0,3), C2H5OH 4(1,4)-3(0,3) and CH3NH2 1(1,0)-1(0,1), CH3OCH3
2(2,1)-2(1,2)EE, C2H5CN 10(4,7)-9(4,6) and H2NCH2CN 10(0,10)-9(3,6) were
presented in Figure 2 and Figure 3. Positions of Sgr B2(N) and Sgr B2(M) are
indicated with crosses. The peak integrated intensity on the Tmb scale are
given in the caption. We could see from Figure 2 and Figure 3 that some
molecular line emission concentrate on Sgr B2(N) and Sgr B2(M), and some
molecular line emission extend to the eastern part of Sgr B2. Based on their
spatial distribution, molecules can be classified as either “extended" that
detected not only in Sgr B2(N) and Sgr B2(M), or “compact" that only detected
toward or near to Sgr B2(N) and Sgr B2(M).

Figure 2: (a) Velocity integrated intensity map of “warm" glycolaldehyde
CH2OHCHO 9(1,8)-8(2,7) (contours) overlayed on velocity integrated intensity
map of “cold" glycolaldehyde CH2OHCHO 1(1,0)-1(0,1) observed at 13 GHz(Li et
al., 2017) in grey scale. The contours are from 3$\sigma$ increasing in steps
of 3$\sigma$, which corresponds to 0.21 K km s-1. The 70 arcsec FWHM ARO beam
at 90 GHz is shown by cyan circle, while the 77 arcsec FWHM TMRT beam at 13
GHz is shown by black circle. (b) Velocity integrated intensity map of CH3OCHO
7(2,5)-6(2,4)E (contours) overlayed on velocity integrated intensity map of
“cold" glycolaldehyde CH2OHCHO 1(1,0)-1(0,1) observed at 13 GHz(Li et al.,
2017) in grey scale. The contours are from 5$\sigma$ increasing in steps of
5$\sigma$, which corresponds to 0.35 K km s-1. The 70 arcsec FWHM ARO beam at
90 GHz is shown by cyan circle, while the 77 arcsec FWHM TMRT beam at 13 GHz
is shown by black circle. (c) Velocity integrated intensity map of t-HCOOH
4(0,4)-3(0,3). The contours are from 5$\sigma$ increasing in steps of
5$\sigma$, which corresponds to 0.35 K km s-1. (d) Velocity integrated
intensity map of C2H5OH 4(1,4)-3(0,3). The red contours started from and with
the step of 1 K km s-1. (e) Velocity integrated intensity map of CH3NH2
1(1,0)-1(0,1). The contours are from 3$\sigma$ increasing in steps of
3$\sigma$, which corresponds to 0.21 K km s-1. The black contours represent
50% of the map peaks, while dots stand for the sampling points in each sub-
plot. The 70 arcsec FWHM ARO beam at 90 GHz is shown by cyan circle in each
sub-plot.

Figure 3: (a) Velocity integrated intensity map of CH3OCH3 2(2,1)-2(1,2)EE
observed with ARO 12m telescope. The contours are from 10$\sigma$ increasing
in steps of 10$\sigma$, which corresponds to 0.7 K km s-1. (b) Velocity
integrated intensity map of C2H5CN 10(4,7)-9(4,6) observed with ARO 12m
telescope. The contours are from 20$\sigma$ increasing in steps of 20$\sigma$,
which corresponds to 1.4 K km s-1. (c) Velocity integrated intensity map of
H2NCH2CN 10(0,10)-9(3,6) observed with ARO 12m telescope. The contours are
from 5$\sigma$ increasing in steps of $\sigma$, which corresponds to 0.04 K km
s-1. The black contours represent 50% of the map peaks. Dots stand for the
sampling points. The 70 arcsec FWHM ARO beam at 90 GHz is shown by cyan circle
in each sub-plot.

The excitation analysis of glycolaldehyde emissions at millimeter and
centimeter wavelengths in Sgr B2(N) indicates that a single temperature
component was unable to reproduce all the observed spectra (Hollis et al.,
2004). Thus, two temperature components were invoked: a “warm" extended
glycolaldehyde envelope surrounded by a “cold" glycolaldehyde halo (Hollis et
al., 2004). The 89 GHz transitions with a higher upper-level energy (26.43 K)
are regarded as coming from the “warm" phase, while transitions with a lower
upper-level energy (1.2 K) are regarded as coming from the “cold" phase. The
“cold" glycolaldehyde was found to be widespread around the Sgr B2 complex (Li
et al., 2017). Figure 2a shows a contour map of the velocity-integrated
intensities of the “warm" glycolaldehyde emission obtained with the ARO 12m
telescope overlaid on the “cold" glycolaldehyde emission acquired with the
Shanghai Tianma 65m telescope in grey scale (Li et al., 2017). The half-power
beamwidths were $\sim$70 arcsec and $\sim$77 arcsec for ARO 12 m observations
and 65m observations, respectively. The “cold" glycolaldehyde is distributed
across 800 arcsec ($\sim$33 pc in diameter), which is substantially more
extended than the “warm" glycolaldehyde. The “warm" glycolaldehyde emission
shows several peaks, including the north cloud, the peak near Sgr B2(M), and
the ridge west of Sgr B2(N). A weak “warm" glycolaldehyde emission at 89 GHz
was also detected toward the eastern component of the Sgr B2 complex.

According to BIMA observations (Hollis et al., 2001), glycolaldehyde is
greatly extended in comparison to the ethyl cyanide and dimethyl ether, which
are largely confined to the Large Molecule Heimat source (LMH) (Snyder, Kuan &
Miao, 1994; Miao et al., 1995), so these molecules were classified as either
“extended" or “compact" here based on the half-power radius of the spatial
distributions in Figures 2 and 3. Figure 4 shows variation of the normalized
integrated intensity of observed transitions with the distance to Sgr B2(N)
for sampling points with $\Delta$RA=0. The normalized intensity is obtained
through dividing the integrated intensity by the maximum integrated intensity
of molecule. We change the declination and take a measurement at constant RA
($\Delta$RA=0). FWHM, the full width at half maximum, is always used to
evaluate the line width of spectral line. Thus we selected 50% as a metric to
evaluate the distribution of molecules. We could see from Figure 4 that the
normalized intensity of CH3OCH3, C2H5CN and H2NCH2CN quickly decrease below
50% for sampling points away from Sgr B2(N). The half-power radius of these
molecules are smaller than the sampling interval (1′). Thus these three
molecules are classified as “compact” molecules. H2NCH2CN was only detected in
Sgr B2(N) in our observations, thus it is also classified as “compact"
molecules. On the other hand, the normalized intensity of CH2OHCHO, CH3OCHO,
t-HCOOH, C2H5OH and CH3NH2 vary slowly as the distance to Sgr B2(N), thus
these molecules are classified as “extended" molecules. In order to
investigate whether only the strongest lines are extended, the “extended" line
profiles were colored with red, while the “compact" line profiles were colored
with blue in spectra of Sgr B2N (Figure 1 upper panel). As is shown in Figure
1 upper panel, the strongest emissions come from C2H5CN, C2H5OH, t-HCOOH and
CH3OCH3. Among them, C2H5CN and CH3OCH3 emission are confined to Sgr B2N and
Sgr B2M. Thus it is sure that not all the strongest lines are the most
extended. It is noted that CH2OHCHO peaks away from Sgr B2N, which seems to
differ from most COMs presented in this paper. The line blending makes it
difficult to obtain the intensity of CH2OHCHO accurately. We have mapped
CH2OHCHO around Sgr B2 with IRAM 30m telescope with better spatial and
spectral resolution to further investigate this issue. The preliminary result
is consistent with ARO 12m result present here. Detailed chemical model is
needed to explain the spatial distribution of glycolaldehyde in this region.

Figure 4: Variation of the normalized intensity of observed transitions versus
the distance to Sgr B2(N) for sampling points with $\Delta$RA=0. The
normalized intensity is obtained through dividing the integrated intensity by
the maximum integrated intensity of molecule. We change declination and take a
measurement at constant RA ($\Delta$RA=0).

###  3.1 “Extended" Molecules

Glycolaldehyde CH2OHCHO: CH2OHCHO, which is a sugar-related molecule, can
react with propenal to form ribose - a central constituent of RNA (Sharma et
al., 2016). CH2OHCHO, was first detected in Sgr B2(N) based on the emission
spectra in the millimeter-wave range observed with the NRAO 12m telescope
(Hollis et al., 2000). The CH2OHCHO 8(0,8)-7(1,7) emission at 89 GHz was
regarded to come from “warm" glycolaldehyde, while the 13 GHz emission was
regarded to come from “cold" glycolaldehyde (Hollis et al., 2004). The
distribution of “warm" glycolaldehyde is extended and seems to be comprised of
several emission cores (see Figure 2). Sensitive interferometric observations
are needed to resolve and study these cores in detail. The “cold"
glycolaldehyde has been found to be widespread around the Sgr B2 complex (Li
et al., 2017). The different spatial distribution of “warm" and “cold"
glycolaldehyde seems to be caused by different excitation conditions,
including temperature and density of molecular gas.

Methyl formate CH3OCHO: ARO observations indicate that the spatial
distribution of methyl formate emission significantly differ from both the
“warm" and the “cold" glycolaldehyde. It seems to peak toward Sgr B2(N) and
Sgr B2(M), and decrease toward the surroundings. Weak emission was seen at the
eastern part of Sgr B2 complex.

Formic acid t-HCOOH: t-HCOOH is a key organic molecule as the carboxyl group
(C(=O)OH) is one of the main functional groups of amino acids (the structural
units of proteins). This species is involved in a chemical route leading to
glycine, the simplest amino acid (Redondo et al., 2015). ARO observations
indicate that the spatial distribution of t-HCOOH extends to the eastern
component and peaks at Sgr B2(N).

Ethanol C2H5OH: C2H5OH was first detected toward Sgr B2 (Zuckerman et al.,
1975). This molecular was found to be present only in the dense ($\geq 10^{6}$
cm-3) and hot ($\geq$100 K) cores associated with newly formed massive stars
(Widicus Weaver et al., 2017; Qin et al., 2010; Bisschop et al., 2007; Martín-
Pintado et al., 2001). Martín-Pintado et al. (2001) found that C2H5OH emission
in Sgr B2 is widespread. They proposed that C2H5OH formed in grains and
released to gas phase by shocks in the last $\sim 10^{5}$ yr. Results present
here provide further evidence for the presence of extended ethanol around Sgr
B2. The C2H5OH emission peaks at the north of Sgr B2(N).

Methylamine CH3NH2: CH3NH2 was first detected in Sgr B2(OH) (Kaifu et al.,
1974). It was also detected in the southern region of Sgr B2 (Halfen et al.,
2013). CH3NH2 is thought to be potential interstellar precursors to the amino
acid glycine, NH2CH2COOH. The reaction of CH3NH2 with CO2 in water ice has
been shown to yield NH2CH2COOH after UV irradiation (Bossa et al., 2009; Lee
et al., 2009). Our observations indicate that CH3NH2 emission peaks at Sgr
B2(N), and extends to the eastern part of Sgr B2 complex, which is in
agreement with previous results.

###  3.2 “Compact" Molecules

Dimethyl ether CH3OCH3: The CH3OCH3 emission is detected toward Sgr B2(N) and
a few points near to Sgr B2(N). Though CH3OCH3 is isomer of C2H5OH, the
distribution of these two molecules are strikingly different, implying that
they have different chemical route, which is consistent with observations in
W51 (Rong et al., 2015). As C2H5OH is thought to be produced by grain-surface
reactions, CH3OCH3 is likely to form via gas-phase chemistry. The spatial
distribution of CH3OCH3 also differ from those of CH3OCHO, which seem to be
different from observations in massive star-forming regions in Galactic disk.
Striking similarity was found in the spatial distribution, temperature and
column densities of these two molecules in star-forming regions in Galactic
disk (Brouillet et al., 2013; Jaber et al., 2014; Rong et al., 2015). CH3OCH3
has been proposed as a precursor molecule to CH3OCHO (Balucani et al., 2015).
The obvious difference in spatial distribution of CH3OCH3 and CH3OCHO observed
in Sgr B2 suggests that the formation mechanism of these two molecules might
differ in the Galactic center and Galactic disk sources.

Ethyl cyanide C2H5CN: Previous observations of C2H5CN in Sgr B2 has shown that
this molecule is confined to a small region, and likely to be located in the
hot, dense core of the star-forming region (Miao & Snyder, 1997). Our mapping
results show that C2H5CN was only detected toward Sgr B2(N), and four points
near to Sgr B2(N), which agree with previous studies.

Amino acetonitrile H2NCH2CN: H2NCH2CN was first detected in Sgr B2(N)
(Belloche et al., 2008). The source size was estimated to be about 2” FWHM. No
evidence for colder, more extended emission was found. Amino acetonitrile may
well be a direct precursor of glycine. In our observations, H2NCH2CN was only
detected toward Sgr B2(N).

###  3.3 Column densities

According to IRAM 30m observations of COMS (Requena-Torres et al., 2006),
$T_{rot}$ derived from C2H5OH is 72.6 K for Sgr B2N, and 55.4 K for Sgr B2M.
$T_{rot}$ derived from C2H5OH range from 9-14 K for other Galactic center
clouds, $T_{rot}$ derived from CH3OCHO range from 8-16 K for other Galactic
center clouds. Assuming the excitation temperature of 50 K for Sgr B2N and Sgr
B2(M), and 14 K for other positions, we calculate the total column densities
of “warm" CH2OHCHO, CH3OCHO, C2H5OH and t-HCOOH for the i, j-th grid in the
position-position space $N_{ij}$ with the expression reported by Hollis et al.
(2004):

$\displaystyle N_{ij}=\frac{3kQe^{E_{u}/kT_{S}}}{8\pi^{3}\nu
S\mu^{2}}\times\frac{\frac{1}{2}\sqrt{\frac{\pi}{ln2}}\frac{\triangle
T_{A}^{*}\triangle
V}{\eta_{B}}}{1-\frac{e^{h\nu/kT_{S}}-1}{e^{h\nu/KT_{bg}}-1}},$ (1)

in which $k$ is the Boltzmann constant in erg K-1, $\frac{\triangle
T_{A}^{*}\triangle V}{\eta_{B}}$ is the observed line integrated intensity in
K Km s-1, $\nu$ is the frequency of the transition in Hz, and $S\mu^{2}$ is
the product of the total torsion-rotational line strength and the square of
the electric dipole moment. $T_{S}$ and $T_{bg}$ (=2.73 K) are the excitation
temperature and background brightness temperature, respectively. $E_{u}/k$ is
the upper level energy in K. The partition function, $Q$, was estimated by
fitting the partition function at different temperatures given in CDMS (Müller
et al., 2005). Values of $E_{u}/k$ and $S\mu^{2}$ are also taken from CDMS.

Because of effects of beam dilution, we did not calculate the column densities
of “compact" molecules like C2H5CN and CH3OCH3. In the below discussion, we
focus only on oxygen-bearing molecules showing extended emission, including
“warm" CH2OHCHO, CH3OCHO, C2H5OH and t-HCOOH.

We made use of the H2 column density obtained with BGPS 1.1 mm data (Bally et
al., 2010) for abundance estimation of molecules. For each position, we
searched for nearest clump identified with BGPS data, and obtained the
corresponding hydrogen column density. The H2 column densities range from
$1.1\times 10^{23}$ cm-2 to $1.0\times 10^{25}$ cm-2 in our observing region,
and peak toward Sgr B2(N) and Sgr B2(M).

##  4 Discussions

A comparison between the abundances of ethanol and “warm" glycolaldehyde
relative to that of $N(H_{2})$ for each position is shown in Figure 5. We
found that $X(CH_{2}OHCHO-warm)=0.2X(C_{2}H_{5}OH)$, with a correlation
coefficient of 0.96. This result agrees well with the relation found for star-
forming regions in the Galactic disk: L1157-b1 (Lefloch et al., 2017), IRAS
16293-2422 (Jørgensen et al., 2016), IRAS 2A and IRAS 4A (Taquet et al., 2015)
(see Figure 3). The strong correlation between ethanol and “warm"
glycolaldehyde suggests that the “warm" glycolaldehyde may be chemically
related to ethanol. Recently, theoretical studies have shown that a new gas-
phase scheme of reactions, involving ethanol as a parent molecule, can lead to
the formation of glycolaldehyde and formic acid (Skouteris et al., 2018). This
model can well explain the abundance correlation between “warm" glycolaldehyde
and ethanol. This model is further supported by the strong correlation between
the abundances of ethanol and formic acid relative to that of $N(H_{2})$
(Figure 5), with a correlation coefficient of 0.92. The correlation between
the abundances relative to H2 of ethanol, “warm" glycolaldehyde and formic
acid suggests that “warm" glycolaldehyde and formic acid are chemically
related to ethanol and are possibly produced by ethanol via gas-phase
reactions (Skouteris et al., 2018). However, we couldnot exclude the
possibility that these molecules are co-spatial in the same gas and trace a
warm gas phase. Interferometric observations of a large sample of interstellar
sources could help to investigate whether ethanol, formic acid and “warm"
glycolaldehyde are co-spatial and constrain their formation pathways (Xue et
al., 2019).

Figure 5: (a) “Warm" glycolaldehyde abundance as a function of ethanol
abundance for different positions in Sgr B2. The relation $X(CH_{2}OHCHO-
warm)=0.2X(C_{2}H_{5}OH)$ is marked by the black dashed line. (b) Formic acid
abundance as a function of ethanol abundance for different positions in Sgr
B2. The relation $X_{t-HCOOH}=0.06X_{C_{2}H_{5}OH}$ is marked by the black
dashed line. (c) “Cold" glycolaldehyde abundance as a function of ethanol
abundance for different positions in Sgr B2. The relation $X_{CH_{2}OHCHO-
cold}=0.94X_{C_{2}H_{5}OH}$ is marked by the black dashed line.

Figure 5 shows a comparison between the abundances relative to H2 of ethanol
and “cold" glycolaldehyde for each position. The abundance of “cold"
glycolaldehyde was observed at the 13 GHz (Li et al., 2017) range from
$10^{-11}$ to $10^{-9}$, which is similar to the value for ethanol. We found
that $X_{CH_{2}OHCHO-cold}=0.94X_{C_{2}H_{5}OH}$, with a correlation
coefficient of 0.98. Theoretical studies show that gas-phase chemistry cannot
produce the observed abundance of ethanol, even under the most favorable
conditions (Charnley et al., 1995). Infrared observations also indicate that
interstellar ethanol can be formed by grain-surface reactions significantly
more efficiently than by gas-phase chemistry (Sandford et al., 1991); thus,
ethanol is thought to be a good tracer of dust chemistry. The good correlation
($\sim$1) between the abundances of ethanol and “cold" glycolaldehyde indicate
that the “cold" glycolaldehyde should also be produced by grain-surface
reactions. Similar to ethanol, the desorption mechanism for the “cold"
glycolaldehyde should be the large-scale shocks in the Galactic center region
(Martín-Pintado et al., 2001).

Laboratory studies show that glycolaldehyde always forms on icy grains at low
temperatures, regardless of UV-photon absorption (i.e. energetic processes),
and methyl formate cannot form efficiently in the absence of energetic
processes (Chuang et al., 2017). Overabundant methyl formate would mean that
additional UV photons to low-temperature surface chemistry. In contrast,
under-abundant methyl formate would mean that only the low-temperature process
is important in the production of complex organic molecules. This view agrees
well with the results presented here, in which methyl formate emission is
concentrated on Sgr B2(N) and Sgr B2(M), where massive star-forming activity
takes place and UV-photons are abundant. In contrast, “cold" glycolaldehyde
emission is very widespread and extends to the eastern part of the Sgr B2
complex, where no star-forming activity has been found and the UV radiation
should be substantially weaker. Thus, the methyl formate and “cold"
glycolaldehyde near to Sgr B2(N) and Sgr B2(M) likely form on icy grains under
UV radiation, while “cold" glycolaldehyde in regions away from the HII region
likely form on icy grains at low temperature.

##  5 SUMMARY

We have carried out large-scale mapping observations of a series of COMs
toward Sgr B2 with the ARO 12m telescope. We obtained large-scale spatial
distribution of COMs, including CH2OHCHO, CH3OCHO, t-HCOOH, C2H5OH and CH3NH2,
CH3OCH3, C2H5CN, in Sgr B2. The main results of this work include:

1\. The spatial distribution of complex organic molecules can be classified as
either “extended" or “compact". The “extended" molecules include CH2OHCHO,
CH3OCHO, t-HCOOH, C2H5OH and CH3NH2, while the “compact" molecules include
CH3OCH3, C2H5CN, and H2NCH2CN.

2\. The spatial distribution of CH3OCHO obviously differ from CH3OCH3, which
have been observed to have similar spatial distribution and column densities
to that in star-forming regions of Galactic disk. These results suggest that
the formation mechanisms of these two molecules in Galactic center may differ
from that in star-forming regions of Galactic disk.

3\. These “compact" molecules likely to be produced under strong UV radiation,
while “extended" molecules likely to be formed under low-temperature, via gas-
phase or grain surface reactions.

4\. We found evidence for an overabundance of CH2OHCHO compared to that
expected from the gas-phase model, which indicates that grain-surface
reactions are necessary to explain the origin of CH2OHCHO in Sagittarius B2.

Our measurements demonstrate the necessity of grain-surface chemistry for the
production of COMs, such as glycolaldehyde in Sgr B2. The results open up an
exciting opportunity to study COMs in the circumnuclear disk of starburst
galaxies with interferometers including ALMA, the Next Generation Very Large
Array (ngVLA) and SKA. Future high-sensitivity interferometric observations
possess the potential to probe COM emissions to investigate whether CMZ clouds
could serve as a template for the nuclei of starburst galaxies in the nearby
and distant universe (Kauffman et al., 2017).

## Acknowledgements

The Kitt Peak 12 Meter is operated by the Arizona Radio Observatory (ARO),
Steward Observatory, University of Arizona. We thank ARO staff for assisting
with the observations. This work is partially supported by the National Key
R&D Program of China (No. 2017YFA0402604), the Natural Science Foundation of
China (11590780, 11590782, 11590784, U1431125, 11773054 and 11903038), the
Special Funding for Advanced Users, budgeted and administrated by Center for
Astronomical Mega-Science, Chinese Academy of Sciences (CAMS-CAS) and CAS
“Light of West China” Program.

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# Bounds for the collapsibility number of a simplicial complex and non-cover
complexes of hypergraphs

Rekha Santhanam111Department of Mathematics, Indian Institute of Technology
Bombay, India<EMAIL_ADDRESS>Samir Shukla222School of Mathematical and
Statistical Sciences, Indian Institute of Technology Mandi, India.
<EMAIL_ADDRESS>Anurag Singh333Department of Mathematics, Indian
Institute of Technology Bhilai, India<EMAIL_ADDRESS>
###### Abstract

The collapsibility number of simplicial complexes was introduced by Wegner in
order to understand the intersection patterns of convex sets. This number also
plays an important role in a variety of Helly type results. There are only a
few upper bounds for the collapsibility number of complexes available in
literature. In general, it is difficult to establish such non-trivial upper
bounds. In this article, we construct a sequence of upper bounds
$\theta_{k}(X)$ for the collapsibility number of a simplicial complex $X$. We
also show that the bound given by $\theta_{k}$ is tight if the underlying
complex is $k$-vertex decomposable. We then give an upper bound for
$\theta_{k}$ and therefore for the collapsibility number of the non-cover
complex of a hypergraph in terms of its covering number.

Keywords : collapsibility, non-cover complexes, hypergraphs

2020 Mathematics Subject Classification: 05C69, 05E45, 52B22, 52A35

##  1 Introduction

Let $X$ be a (finite) simplicial complex. Let $\gamma,\sigma\in X$ be such
that $|\gamma|\leq d$ and $\sigma\in X$ is the only maximal simplex that
contains $\gamma$. Then, $(\gamma,\sigma)$ is called a free pair and $\gamma$
is called a free face of $\sigma$ in $X$. An elementary $d$-collapse of $X$ is
the simplicial complex $X^{\prime}$ obtained from $X$ by removing all those
simplices $\tau$ of $X$ such that $\gamma\subseteq\tau\subseteq\sigma$, and we
denote this elementary $d$-collapse by $X\xrightarrow{\gamma}X^{\prime}$. The
complex $X$ is called _$d$ -collapsible_ if there exists a sequence of
elementary $d$-collapses

$X=X_{1}\xrightarrow{\gamma_{1}}X_{2}\xrightarrow{\gamma_{2}}\cdots\xrightarrow{\gamma_{k-1}}X_{k}=\emptyset$

from $X$ to the empty complex $\emptyset$. Note that every $d$-dimensional
complex is always $d+1$ collapsible. Clearly, if $X$ is $d$-collapsible and
$d<c$, then $X$ is $c$-collapsible. The _collapsibility number_ of $X$,
denoted as $\mathcal{C}(X)$, is the minimal integer $d$ such that $X$ is
$d$-collapsible.

The notion of d-collapsibility of simplicial complexes was introduced by
Wegner [16]. The motivation of introducing d-collapsibility comes from
combinatorial geometry as a tool for studying intersection patterns of convex
sets [6, 9, 15, 16]. The collapsibility number plays an important role in the
study of various Helly type results [1, 8, 9]. The collapsibility number is
also related to an another well-studied combinatorial inviariant of a
simplicial complex called Leray number $\mathcal{L}(X)$ (Definition 2.14). In
[16], Wegner proved that $\mathcal{L}(X)\leq\mathcal{C}(X)$ for any simplicial
complex $X$.

In general, it is difficult to determine the collpasibility number or bounds
for the collapsibility number of a simplicial complex. In [2], Biyikoğlu and
Civan, introduced a combinatorial invariant $\theta(X)$ for a simplicial
complex $X$ and proved that $\mathcal{C}(X)\leq\theta(X)$ always holds. They
also gave an example of a complex $X$, where $\mathcal{C}(X)\neq\theta(X)$.
The main motivation behind this project is to find an invariant for a
simplicial complex which is a better approximation to the collapsibility
number than the theta number. In this article, we introduce a new
combinatorial invariant $\theta_{k}(X)$ for a simplicial complex $X$ and each
non-negative integer $k$, and show that
$\mathcal{C}(X)\leq\theta_{k}(X)\leq\theta_{k-1}(X)\leq\ldots\leq\theta_{1}(X)\leq\theta_{0}(X)=\theta(X)$
(see Remark 2.3 and Theorem 2.5). We also give an example of a complex $X$,
where $\mathcal{C}(X)=\theta_{1}(X)<\theta(X)$ (see Example 2.4). Further, we
prove that for $k$-vertex decomposable simplical complexes $X$,
$\mathcal{C}(X)=\theta_{k}(X)$ (see Theorem 2.9).

One of the most commonly used method of establishing an upper bound on the
collapsibility number of a complex $X$ is by computing the number $d(X,\prec)$
which is given by the minimal exclusion sequence of simplices of $X$ (see
Proposition 3.2 for more details). In this article, we prove that $d(X,\prec)$
is in fact an upper bound for $\theta_{0}(X)$ (which can be strictly greater
than $\mathcal{C}(X)$ for several complexes) and therefore for all
$\theta_{k}$ for any simplicial complex $X$.

In [4, Theorem 3], the authors showed that the collapsibility number of non-
cover complex of a graph $G$ (denoted $\mathrm{NC}(G)$) is bounded by
$|V(G)|-i_{\gamma}(G)-1$ by computing $d(\mathrm{NC}(G),\prec)$, where
$i_{\gamma}(G)$ denotes the independence domination number of $G$. In Theorem
3.1, we prove that for any hypergraph $\mathcal{H}$,
$\theta_{0}(\mathrm{NC}(\mathcal{H}))$ (and therefore
$\mathcal{C}(\mathrm{NC}(\mathcal{H}))$) is bounded above by the
$|V(\mathcal{H})|-\tau(\mathcal{H})-1$, where $\tau(\mathcal{H})$ (defined in
Section 3) is the covering number of $\mathcal{H}$.

In [11], authors defined three invariants, strong total domination number
$\tilde{\gamma}(\mathcal{H})$, strong independence domination number
$\gamma_{si}(\mathcal{H})$ and edgewise-domination number
$\gamma_{E}(\mathcal{H})$ for a hypergraph $\mathcal{H}$ (see Definitions 3.7,
3.8 and 3.9). Under suitable conditions on $\mathcal{H}$, they showed that
$\mathcal{L}(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\max\\{\left\lceil{\tilde{\gamma}(\mathcal{H})}/{2}\right\rceil,\gamma_{si}(\mathcal{H}),\gamma_{E}(\mathcal{H})\\}-1$.
We give a class of hypergraphs (see Example 3.11) for which

$\tau(\mathcal{H})>\max\\{\tilde{\gamma}(\mathcal{H}),\gamma_{si}(\mathcal{H}),\gamma_{E}(\mathcal{H})\\}.$

Since $\mathcal{L}(X)\leq\mathcal{C}(X)$ for any simplicial complex $X$, our
result, Theorem 3.1 implies a more general result,

$\mathcal{L}(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\max\\{\tau(\mathcal{H}),\left\lceil\frac{\tilde{\gamma}(\mathcal{H})}{2}\right\rceil,\gamma_{si}(\mathcal{H}),\gamma_{E}(\mathcal{H})\\}-1.$

##  2 The $\theta_{k}$ number of a complex

Tancer [14] showed that the collapsibility number of a simplicial complex is
bounded by the collapsibility number of link and deletion of $X$ with respect
to any vertex $v$. This allows for inductive arguments to find the bounds on
collapsibility number of a simplicial complex [12].

For any simplicial complex $X$ and $\sigma\in X$, the subcomplexes _link_ and
_deletion_ of $\sigma$ in $X$ are defined as follows

$\begin{split}\mathrm{lk}(\sigma,X)&=\\{\tau\in
X:\sigma\cap\tau=\emptyset,~{}\sigma\cup\tau\in X\\},\\\
\mathrm{del}(\sigma,X)&=\\{\tau\in X:\sigma\nsubseteq\tau\\}.\end{split}$

Biyikoğlu and Civan [2] defined $\theta(X)$ inductively along these ideas for
any simplicial complex $X$. Let $X^{o}$ denote the set of vertices $v$ in $X$
such that $\mathrm{lk}(v,X)\neq\mathrm{del}(v,X)$. If $X^{o}$ is empty then
$\theta(X)=0$, else

$\theta(X)=\min_{v\in
X^{0}}\\{\max\\{\theta(\mathrm{lk}(v,X))+1,\theta(\mathrm{del}(v,X)\\}\\}.$

They showed that $\mathcal{C}(X)\leq\theta(X)$ for all $X$ and proved several
properties for $\theta(X)$. We expand on their ideas and define a sequence of
invariants $\theta_{k}(X)$ which lie between $\mathcal{C}(X)$ and $\theta(X)$.

We first introduce the notation we use throughout this paper. Let $X$ be a
simplicial complex. We denote the set of vertices of $X$ by $V(X)$. For
$A\subseteq V(X)$, the induced subcomplex on vertex set $A$ is
$X[A]=\\{\sigma\in X:\sigma\subseteq A\\}$. For $k\geq 0$, we let $X_{(k)}$
denote the set of $k$-dimensional faces of $X$ and

$X_{(k)}^{o}=\\{\sigma\in X_{(k)}:\mathrm{lk}{(\sigma,X)}\neq X[(V(X)\setminus
V(\sigma)]\\}.$

######  Lemma 2.1.

Let $X$ be a simplicial complex of dimension at least $k$. If
$X_{(k)}^{o}=\emptyset$, then $X$ is a simplex.

###### Proof.

If $k=0$, then we prove the result by the induction on the number of vertices
of $X$. The base case (i.e., $X$ is a vertex) is trivially true. Now observe
that, if $X_{(0)}^{o}=\emptyset$ then $\mathrm{lk}(v,X)_{(0)}^{o}=\emptyset$.
Moreover, for any vertex $v$, $\mathrm{lk}(v,X)=X-\\{v\\}$, which implies that
$X=(X-\\{v\\})\ast\\{v\\}$. Hence by induction on the number of vertices, we
get that $\mathrm{lk}(v,X)$ is simplex and therefore $X$ is a simplex.

Let $k>0$. Let $\sigma\in X$ be a $k$-dimensional simplex. Then
$\mathrm{lk}(\sigma,X)=X[V(X)\setminus\sigma]$. Hence
$X=\mathrm{lk}(\sigma,X)\ast\sigma$. Let $Y=\mathrm{lk}(\sigma,X)$. If
$Y_{0}^{o}=\emptyset$, then $Y$ is a simplex and therefore $X$ is a simplex.
If $Y_{(0)}^{o}\neq\emptyset$, then $\mathrm{lk}(v,Y)\neq Y-\\{v\\}$ for some
$v\in V(Y)$. Choose $w\in\sigma$ and let
$\tau=(\sigma\setminus\\{w\\})\cup\\{v\\}$. Then $\mathrm{lk}(\tau,X)\neq
X[V(X)\setminus\tau]$, a contradiction. Hence $Y_{(0)}^{o}=\emptyset$. By
induction $Y$ is a simplex and therefore $X=Y\ast\sigma$ implies that $X$ is a
simplex. ∎

######  Definition 2.2.

Define $\theta_{0}(X)=\theta^{\prime}_{0}(X)=\theta(X)$ and for $k\geq 1$,
define $\theta_{k}$ inductively as follows.

$\theta_{k}^{\prime}(X)=\begin{cases}\theta_{k-1}(X)&\text{if
}X_{(k)}^{o}=\emptyset,\\\ \min\limits_{\sigma\in
X_{(k)}^{o}}\\{\max\\{\theta^{\prime}_{k}(\mathrm{del}{(\sigma,X))},\theta^{\prime}_{k}(\mathrm{lk}{(\sigma,X))}+k+1\\}&\text{otherwise},\end{cases}$

and $\theta_{k}(X)=\min\\{\theta^{\prime}_{k}(X),\theta_{k-1}(X)\\}$.

######  Remark 2.3.

Note by definition $\theta_{k}(X)\leq\theta_{k-1}(X)$ for all $k\geq 1$.

We now give an example where $\theta_{1}<\theta_{0}$.

######  Example 2.4.

(Example V6F10-6 from [13]) Let $\Delta$ be the simplicial complex on the
vertex set $\\{1,2,3,4,5,6\\}$ with the set of facets

$\\{\\{1,2,3\\},\\{1,2,4\\},\\{1,2,5\\},\\{1,3,4\\},\\{1,3,6\\},\\{2,4,5\\},\\{2,5,6\\},\\{3,4,6\\},\\{3,5,6\\},\\{4,5,6\\}\\}.$

This example was also discussed in [5] as an example of a complex which is
$1$-vertex decomposable but not $0$-vertex decomposable. Thus, from Theorem
2.9, we get that $\mathcal{C}(\Delta)=\theta_{1}(\Delta)$. We now show that
$\theta_{1}(\Delta)\leq 2<3\leq\theta_{0}(\Delta)$.

To compute $\theta_{1}(\Delta)$, let us look at
$\theta_{1}(\mathrm{lk}(\\{1,5\\},\Delta))$ and
$\theta_{1}(\mathrm{del}(\\{1,5\\},\Delta))$. Observe that
$\mathrm{lk}(\\{1,5\\},\Delta)$ is a point $\\{2\\}$ implying that
$\theta_{1}(\mathrm{lk}(\\{1,5\\},\Delta))=0$. Thus,

$\theta_{1}(\Delta)\leq\max\\{\theta_{1}(\mathrm{del}(\\{1,5\\},\Delta)),2\\}.$

Here, the set of facets of $\mathrm{del}(\\{1,5\\},\Delta))$ is

$\\{\\{1,2,3\\},\\{1,2,4\\},\\{1,3,4\\},\\{1,3,6\\},\\{2,4,5\\},\\{2,5,6\\},\\{3,4,6\\},\\{3,5,6\\},\\{4,5,6\\}\\}.$

It is easy to verify by doing a similar calculation on the deletion complexes
using the sequence
$\\{\\{1,6\\},\\{2,3\\},\\{1,3\\},\\{1,2\\},\\{2,4\\},\\{2,5\\},\\{3,4\\},\\{3,5\\},\\{4,5\\}\\}$
of $1$-faces, link complex at every step is a simplex and the deletion complex
at the end is $1$-dimensional. Observe that $\mathcal{C}(X)$ is always less
than or equal to the dimension of $X$. Hence by using the sequence
$\\{\\{1,6\\},\\{2,3\\},\\{1,3\\},\\{1,2\\},\\{2,4\\},\\{2,5\\},\\{3,4\\},\\{3,5\\},\\{4,5\\}\\}$
of $1$-faces in the $\mathrm{del}(\\{1,5\\},\Delta)$, we conclude that
$\theta_{1}(\Delta)\leq 2$.

Observe that the link of every vertex contains an induced subcomplex
isomorphic to a triangulation of a circle. Hence the collapsibility number of
link of every vertex is $2$. Thus for each vertex $v\in\Delta$,
$\theta_{0}(\mathrm{lk}(v,\Delta))\geq\mathcal{C}(\mathrm{lk}(v,\Delta))\geq
2$. Hence by definition $\theta_{0}(\Delta)\geq 3$.

######  Theorem 2.5.

Let $X$ be a simplicial complexes. Then for any $k\geq 0$,

$\mathcal{C}(X)\leq\theta_{k}(X).$

###### Proof.

Proof is by induction on $k$. From [2, Theorem 25],
$\mathcal{C}(X)\leq\theta_{0}(X)=\theta(X)$. Let $k\geq 1$ and assume that
$\mathcal{C}(X)\leq\theta_{r}(X)$ for $0\leq r<k$. We now prove that
$\mathcal{C}(X)\leq\theta_{k}(X)$ by induction on the number of $k$-simplices
of $X$. If $X$ has no $k$-simplex then clearly $X_{(k)}^{o}=\emptyset$
implying that $\theta_{k}(X)=\theta_{k-1}(X)$ and hence the result follows.

By definition,
$\theta_{k}(X)=\min\\{\theta_{k-1}(X),\theta_{k}^{\prime}(X)\\}$. If
$\theta_{k}(X)=\theta_{k-1}(X)$ then the result follows from induction. Now
assume that $\theta_{k}(X)=\theta_{k}^{\prime}(X)$.

We first prove the following claim.

######  Claim 2.6.

For any $\sigma\in X_{(k)}$ (i.e., $\sigma$ is a $k$-face)

$\mathcal{C}(X)\leq\max\\{\mathcal{C}(\mathrm{del}(\sigma,X),\mathcal{C}(\mathrm{lk}(\sigma,X))+k+1\\}.$

###### Proof of 2.6.

Let $\mathrm{lk}(\sigma,X)$ be $d$-collapsible. Then there exist a sequence of
elementary $d$-collapses such that

$\mathrm{lk}(\sigma,X)=X_{0}\xrightarrow{\sigma_{1}}X_{1}\xrightarrow{\sigma_{2}}X_{2}\ldots\xrightarrow{\sigma_{r}}X_{r}=\emptyset.$

Since $\mathrm{lk}(\sigma,X)\xrightarrow{\sigma_{1}}X_{1}$ is an elementary
collapse, there exist a facet $\tau_{1}\in\mathrm{lk}(\sigma,X)$ such that
$\sigma_{1}$ is a free face of $\tau_{1}$ in $\mathrm{lk}(\sigma,X)$.
Therefore, $\sigma_{1}\cup\sigma$ is a free face of $\tau_{1}\cup\sigma$ in
$X$. Furthermore, since $|\sigma_{1}\cup\sigma|\leq d+k+1$, we get an
elementary $(d+k+1)$-collapse in $X$. Hence, the sequence

$X=Y_{0}\xrightarrow{\sigma_{1}\cup\sigma}Y_{1}\xrightarrow{\sigma_{2}\cup\sigma}Y_{2}\ldots\xrightarrow{\sigma_{r}\cup\sigma}Y_{r}=\mathrm{del}(\sigma,X)$

gives a sequence of elementary $(d+k+1)$-collapses of $X$ onto
$\mathrm{del}(\sigma,X)$. This implies that the collapsibility number of $X$
is less than or equal to
$\max\\{\mathcal{C}(\mathrm{del}(\sigma,X)),d+k+1\\}$. ∎

If $X_{(k)}^{o}=\emptyset$, then the result follows from the induction on $k$
(since $\theta_{k}(X)=\theta_{k-1}(X)$). For $X_{(k)}^{o}\neq\emptyset$, let
$\sigma\in X_{(k)}$ such that
$\theta_{k}^{\prime}(X)=\max\\{\theta_{k}^{\prime}(\mathrm{del}(\sigma,X)),\theta_{k}^{\prime}(\mathrm{lk}(\sigma,X))+k+1\\}$.
From the previous claim, we have that

$\begin{split}\mathcal{C}(X)&\leq\max\\{\mathcal{C}(\mathrm{del}(\sigma,X),\mathcal{C}(\mathrm{lk}(\sigma,X))+k+1\\}\\\
&\leq\max\\{\theta_{k}(\mathrm{del}(\sigma,X)),\theta_{k}(\mathrm{lk}(\sigma,X))+k+1\\}\\\
&\leq\max\\{\theta_{k}^{\prime}(\mathrm{del}(\sigma,X)),\theta_{k}^{\prime}(\mathrm{lk}(\sigma,X))+k+1\\}\\\
&=\theta_{k}^{\prime}(X)=\theta_{k}(X).\end{split}$

Here, the second inequality follows from induction, and the third inequality
follows from the fact that $\theta_{k}(X)\leq\theta_{k}^{\prime}(X)$. ∎

######  Remark 2.7.

Note that Theorem 2.5, along with Example 2.4, implies that $\theta_{1}$ is a
better approximation to $\mathcal{C}(X)$ than $\theta_{0}$.

In our next result, we show that the bound obtained in Theorem 2.5 is tight
for a particular class of complexes known as $k$-vertex decomposable
complexes. Given a simplicial complex $X$ its _pure $n$-skeleton_, $X^{[n]}$
is the subcomplex of $X$ spanned by all $n$-faces of $X$. The complex $X$ is
said to be _pure_ $n$-dimensional complex if $X=X^{[n]}$.

A pure $d$-dimensional simplicial complex $X$ is said to be shellable, if its
maximal simplices can be ordered $\Gamma_{1},\Gamma_{2}\ldots,\Gamma_{t}$ in
such a way that the subcomplex
$(\bigcup\limits_{i=1}^{k-1}\Gamma_{i})\cap\Gamma_{k}$ is pure and
$(d-1)$-dimensional for all $k=2,\ldots,t$. A pure simplicial complex $X$ is
said to be _Cohen Macaulay_ if, for all simplices $\sigma\in X$, the complex
$\mathrm{lk}(\sigma,X)$ is homologically
($\text{dim}(\mathrm{lk}(\sigma,X))-1$)-connected, i.e.,
$\tilde{H}_{i}(\mathrm{lk}(\sigma,X))=0$ for all
$i<\text{dim}(\mathrm{lk}(\sigma,X)$. As a consequence, we get that if $X$ is
Cohen Macaulay, then $\mathrm{lk}(\sigma,X)$ is also Cohen Macaulay for any
$\sigma\in X$.

Alternatively, a pure simplicial complex $X$ is said to be _Cohen Macaulay_ if
each induced subcomplex $A$ of $X$ is homologically
($\text{dim}(A)-1$)-connected. From this definition and standard facts on
homology, it can be easily verified that if $X$ is Cohen Macaulay, then any
skeleton of $X$ is also Cohen Macaulay.

######  Definition 2.8.

[5, Definition 5.1] For $k\geq 0$, a pure $r$-dimensional simplicial complex
$X$ is said to be $k$-vertex decomposable if $X$ is a simplex or $X$ contains
a face $\sigma$ such that

  1. 1.

$\text{dim}(\sigma)\leq k$.

  2. 2.

both $\mathrm{del}(\sigma,X)$ and $\mathrm{lk}(\sigma,X)$ are $k$-vertex
decomposable, and

  3. 3.

$\mathrm{del}(\sigma,X)$ is pure and the dimension is same as that of $X$.
(Such a face $\sigma$ is called a shedding face of $X$).

The $k$-vertex decomposability ($k\geq 1$) of a complex interpolates between
the shellability and $0$-vertex decomposability of the complex. More
precisely,

$0\text{-vertex decomp.}\implies k\text{-vertex
decomp.}\implies\text{shellability}\implies\text{Cohen Macaulay}.$

The first two implications are discussed in [5, Section 5]. The last
implication follows from [3, Section 11].

######  Theorem 2.9.

If $X$ is $k$-vertex decomposable for some $k\geq 0$, then

$\mathcal{C}(X)=\theta_{k}(X)=\theta^{\prime}_{k}(X).$

Note that the authors in [2] showed that $\mathcal{C}(X)=\theta_{0}(X)$
whenever $X$ is vertex decomposable complex, i.e., 0-vertex decomposable. To
prove Theorem 2.9, we need a few results which we prove now.

Recall that $X_{(k)}^{o}=\\{\sigma\in X_{(k)}:\mathrm{lk}{(\sigma,X)}\neq
X[(V(X)\setminus V(\sigma)]\\}$.

######  Lemma 2.10.

Let $X$ be of a $k$-vertex decomposable simplicial complex of dimension at
least $k$. Then $X_{(k)}^{o}\neq\emptyset$.

###### Proof.

Let $X$ be $r$-dimensional, where $r\geq k$. Since $X$ is $k$-vertex
decomposable, there exists a shedding face $\tau\in X$ such that
$\dim(\tau)\leq k$ and dim$(\mathrm{del}(\tau,X))=r$. Since $X$ is pure and
$r\geq k$, there exists a $k$-face $\sigma\in X$ such that
$\tau\subseteq\sigma$. We now prove that $\sigma\in X_{(k)}^{o}$. On contrary,
assume that $\sigma\notin X_{(k)}^{o}$, i.e.,
$\mathrm{lk}{(\sigma,X)}=X[(V(X)\setminus V(\sigma)]$. Let $\gamma$ be a facet
of $X[(V(X)\setminus V(\sigma)]$. This implies that $\gamma\sqcup\sigma$ is a
facet of $X$. Hence, $(\gamma\cup\sigma)\setminus\tau$ is a facet of
$\mathrm{del}(\tau,X)$. Now observe that
dim$((\gamma\cup\sigma)\setminus\tau)<\dim(\gamma\cup\sigma)=r$. This
contradicts the fact that $\mathrm{del}(\tau,X)$ is pure and of dimension $r$.
Hence $\sigma\in X_{(k)}^{o}$. ∎

The following proposition is a generalization of [7, Theorem 4.2].

######  Lemma 2.11.

Let $Y$ be a simplicial complex and suppose that
$\tilde{H}_{n-k}(\mathrm{lk}(\sigma,Y))\neq 0$ for a $k$-face $\sigma\in Y$.
If $\mathrm{lk}(\sigma,Y)^{[n-k]}$ is contained in a subcomplex $Y_{0}$ of
$\mathrm{del}(\sigma,Y)$ with $\tilde{H}_{n-k}(Y_{0})=0$, then
$\tilde{H}_{n+1}(Y)\neq 0$.

###### Proof.

Given a $k$-face $\sigma$, let st$(\sigma,Y)=\\{\tau\in
Y:\sigma\subseteq\tau\\}$. Then,
$Y=\mathrm{del}(\sigma,Y)\cup\text{st}(\sigma,Y)$ and
$\mathrm{del}(\sigma,Y)\cap\text{st}(\sigma,Y)=\mathrm{lk}(\sigma,Y)\ast\partial(\sigma)$,
where $\partial(\sigma)=\\{\tau:\tau\subsetneq\sigma\\}$.

Using Mayer-Vietoris we get that the sequence

$\tilde{H}_{n+1}(Y)\to\tilde{H}_{n}(\mathrm{lk}(\sigma,Y)\ast\partial(\sigma))\xrightarrow{i}\tilde{H}_{n}(\mathrm{del}(\sigma,X))$

is exact. Note that,

$\tilde{H}_{n}(\mathrm{lk}(\sigma,Y)\ast\partial(\sigma))\cong\tilde{H}_{n}(\Sigma^{k}(\mathrm{lk}(\sigma,Y)))\cong\tilde{H}_{n-k}(\mathrm{lk}(\sigma,Y)).$

Since the map $i$ is induced by an inclusion of
$\mathrm{lk}(\sigma,Y)^{[n-k]}$ in $Y_{0}\subseteq\mathrm{del}(\sigma,Y)$, the
map $i$ is trivial. Thus, by the exactness of the above diagram, we get the
required result. ∎

######  Lemma 2.12.

Let $k\geq 1$ be a positive integer, and $Y$ be a simplicial complex. If
$\tau$ is a shedding $k$-face for $Y$, $\mathrm{del}(\tau,Y)$ is Cohen
Macaulay and $\tilde{H}_{n-k}(\mathrm{lk}(\tau,Y))\neq 0$, then
$\tilde{H}_{n+1}(Y)\neq 0$.

###### Proof.

If $\tau$ is a shedding $k$-face for $Y$, then $\mathrm{del}(\tau,Y)$ is a
pure complex of $\text{dim}(Y)$. Moreover, $\mathrm{lk}(\tau,Y)^{[n-k]}$ will
be contained in the subcomplex
$\mathrm{del}(\tau,Y)^{[n-k]}\subseteq\mathrm{del}(\tau,Y)^{[n+1]}$. Further,
$\mathrm{del}(\tau,Y)$ is a Cohen Macaulay complex implies that
$\mathrm{del}(\tau,Y)^{[n]}$ is Cohen Macaulay for all $n\geq 1$. In
particular, choosing $\emptyset=\sigma\in\mathrm{del}(\tau,Y)$ we get that
$\mathrm{del}(\tau,Y)^{[n+1]}=\mathrm{lk}(\sigma,\mathrm{del}(\tau,Y)^{[n+1]})$
is homologically $n$-connected. Therefore, $\tilde{H}_{n+1}(Y)$ $\neq 0$ by
Lemma 2.11. ∎

Our next result establishes the commutativity of link and deletion of disjoint
faces in a complex.

######  Lemma 2.13.

Let $\sigma,\tau\in X$ such that $\sigma\cap\tau=\emptyset$. Then,
$\mathrm{lk}(\tau,(\mathrm{del}(\sigma,X))=\mathrm{del}(\sigma,\mathrm{lk}(\tau,X))$.

###### Proof.

Let $\gamma\in\mathrm{lk}((\tau,\mathrm{del}(\sigma,X))$. Thus
$\gamma\cup\tau\in\mathrm{del}(\sigma,X)$ implying that
$\sigma\nsubseteq(\gamma\cup\tau)$. This gives us that
$\sigma\nsubseteq\gamma$. Moreover, we know that
$\gamma\in\mathrm{lk}(\tau,\mathrm{del}(\sigma,X))\subseteq\mathrm{lk}(\tau,X)$.
Therefore, the last two statements imply that
$\gamma\in\mathrm{del}(\sigma,\mathrm{lk}(\tau,x))$.

Now let $\eta\in\mathrm{del}(\sigma,\mathrm{lk}(\tau,X))$. So,
$\sigma\nsubseteq\eta$. Furthermore, $\sigma\cap\tau=\emptyset$ implies that
$\sigma\nsubseteq\eta\cup\tau$. Hence $\eta\cup\tau\in\mathrm{del}(\sigma,X)$
which gives us that $\eta\in\mathrm{lk}(\tau,\mathrm{del}(\sigma,X))$. ∎

######  Definition 2.14.

A simplicial complex $X$ is called $k$-Leray if $\tilde{H}_{i}(\mathsf{L})=0$
for all $i\geq k$ and for every induced subcomplex $\mathsf{L}\subseteq X$.
The Leray number $\mathcal{L}(X)$ of $X$ is the least integer $k$ for which
$X$ is $k$-Leray.

######  Proposition 2.15.

If $\sigma$ is a shedding $k$-face for a simplicial complex $X$ such that
$\mathrm{del}(\sigma,X)$ is Cohen-Macaulay, then
$\mathcal{L}(X)\geq\max\\{\mathcal{L}(\mathrm{del}(\sigma,X)),\mathcal{L}(\mathrm{lk}(\sigma,X))+k+1\\}$.

###### Proof.

By [7, Lemma 2.3], $\mathcal{L}(X)\geq d$ if and only if
$\tilde{\text{H}}_{d-1}(\mathrm{lk}(\gamma,X))\neq 0$ for some $\gamma\in X$.
Let $\mathcal{L}(\mathrm{lk}(\sigma,X))=d$, then there exists a face
$\tau\in\mathrm{lk}(\sigma,X)$ such that
$\tilde{H}_{d-1}(\mathrm{lk}(\tau,\mathrm{lk}(\sigma,X)))\neq 0$.

Since $\tau\cap\sigma=\emptyset$, by Lemma 2.13,
$\mathrm{lk}(\tau,(\mathrm{del}(\sigma,X))=\mathrm{del}(\sigma,\mathrm{lk}(\tau,X))$.
Furthermore, since the link of any simplex in a Cohen-Macaulay complex is
again Cohen-Macaulay, the complex
$\mathrm{lk}(\tau,(\mathrm{del}(\sigma,X))=\mathrm{del}(\sigma,\mathrm{lk}(\tau,X))$
is Cohen-Macaulay. Since the link of any face in a pure complex is again pure,
it is easy to check that $\sigma$ is a shedding face for $\mathrm{lk}(\tau,X)$
as well. Since $\tilde{H}_{d-1}(\mathrm{lk}(\sigma,\mathrm{lk}(\tau,X))\neq
0$, by Lemma 2.12, we get that $\tilde{H}_{d+k}(\mathrm{lk}(\tau,X))\neq 0$.
This implies that $\mathcal{L}(X)\geq d+k+1$.

Now it is sufficient to prove that
$\mathcal{L}(X)\geq\mathcal{L}(\mathrm{del}(\sigma,X))$. The proof is by
induction on number of vertices in $X$. Let $Y=\mathrm{del}(\sigma,X)$. Let
$A\subseteq V(Y)$. Then observe that $Y[A]=\mathrm{del}(\sigma\cap A,X[A])$.
By induction, $\mathcal{L}(X[A])\geq\mathcal{L}(Y[A])$. Since
$\mathcal{L}(X)\geq\mathcal{L}(X[A])$ by taking $A=V(Y)$, we get that

$\mathcal{L}(X)\geq\mathcal{L}(X[A])\geq\mathcal{L}(Y[A])=\mathcal{L}(Y).$

∎

We can now prove Theorem 2.9.

###### Proof of Theorem 2.9.

We know that
$\mathcal{L}(X)\leq\mathcal{C}(X)\leq\theta_{k}(X)\leq\theta_{k}^{\prime}(X)$.
We will now prove that $\theta_{k}^{\prime}(X)\leq\mathcal{L}(X)$ by induction
on the number of $k$-faces of $X$. The base case is when the complex has only
one $k$-face, i.e., the complex is a simplex. In this case
$\mathcal{L}(X)=0=\theta_{k}^{\prime}(X)$. Since $X$ is $k$-vertex
decomposable, Lemma 2.10 implies that $X_{(k)}^{o}\neq\emptyset$ and any
shedding $k$-face is in $X_{(k)}^{o}$. Also, since $X$ is $k$-vertex
decomposable there exists a $k$-dimensional shedding face $\sigma$ of $X$ such
that $\sigma\in X_{(k)}^{o}$ and $\mathrm{del}(\sigma,X)$ is a pure $k$-vertex
decomposable complex and therefore Cohen-Macaulay. From Proposition 2.15 we
get that
$\mathcal{L}(X)\geq\max\\{\mathcal{L}(\mathrm{del}(\sigma,X),\mathcal{L}(\mathrm{lk}(\sigma,X)+k+1\\}$.
Thus, from Definition 2.2, we have that

$\begin{split}\theta_{k}^{\prime}(X)&\leq\max\\{\theta_{k}^{\prime}(\mathrm{del}(\sigma,X)),\theta_{k}^{\prime}(\mathrm{lk}(\sigma,X))+k+1\\}\\\
&\leq\max\\{\mathcal{L}(\mathrm{del}(\sigma,X)),\mathcal{L}(\mathrm{lk}(\sigma,X))+k+1\\}\\\
&\leq\mathcal{L}(X).\end{split}$

Here, the second inequality follows from induction. ∎

The following can be easily inferred from Theorem 2.9 and the fact that
$k$-vertex decomposability implies shellability.

######  Remark 2.16.

If $X$ is $k$-dimensional pure complex and $\theta_{k}(X)\neq\mathcal{C}(X)$,
then $X$ is not shellable.

##  3 Non-cover complexes of hypergraphs

A hypergraph $\mathcal{H}$ is an ordered pair
$(V(\mathcal{H}),E(\mathcal{H}))$, where $V$ is a (finite) set and $E$ is a
family of subsets of $V$. The elements of $V(\mathcal{H})$ are called the
vertices of $\mathcal{H}$, and the elements of $E(\mathcal{H})$ its edges. Let
$\mathcal{H}$ be a hypergraph. Let $v\in V(\mathcal{H})$. A vertex $w\in
V(\mathcal{H})$ is a neighbour of $v$, if there exists $e\in E(\mathcal{H})$
such that $\\{v,w\\}\subseteq e$. The neighbour set of $v$ is defined as
$N(v)=\\{w:w\text{ is a neighbour of }v\\}$. For $A\subseteq V(\mathcal{H})$,
the neighbour set of $A$ is $N(A):=\bigcup_{v\in A}N(v)$. For $S\subseteq
V(\mathcal{H})$, the induced subgraph $\mathcal{H}[S]$ is the hypergraph on
the vertex set $S$ and any $e\subseteq S$ is an edge in $\mathcal{H}[S]$ if
$e\in E(\mathcal{H})$.

A set $B\subseteq V(\mathcal{H})$ is called a cover of $\mathcal{H}$ if $e\cap
B\neq\emptyset$ for all $e\in E(\mathcal{H})$. The covering number of
$\mathcal{H}$ is defined as

$\tau(\mathcal{H})=\min\\{|B|:B\text{ is a cover of }\mathcal{H}\\}.$

A set $A\subseteq V(\mathcal{H})$ is called a non-cover if it is not a cover
of $\mathcal{H}$. The non-cover complex $\mathrm{NC}(\mathcal{H})$ of
$\mathcal{H}$ is a simplicial complex defined as

$\mathrm{NC}(\mathcal{H})=\\{A\subseteq V(\mathcal{H}):A\ \text{is a non-cover
of}\ \mathcal{H}\\}.$

######  Theorem 3.1.

Let $\mathcal{H}$ be a hypergraph. Then

$C(\mathrm{NC}(\mathcal{H}))\leq\theta_{0}(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\tau(\mathcal{H})-1.$

Before proving Theorem 3.1, we first review the minimal exclusion principle,
which will play a key role in the proof of Theorem 3.1.

Let $X$ be a simplicial complex on a linearly ordered vertex set $V$ and let
$\prec:\gamma_{1},\ldots,\gamma_{m}$ be a linear ordering on the maximal
simplices of $X$. Given a $\gamma\in X$, the minimal exclusion sequence
$\textrm{mes}(\gamma,\prec)$ defined as follows:

Let $j$ denote the smallest index such that $\gamma\subseteq\gamma_{j}$. If
$j=1$, then $\textrm{mes}(\gamma,\prec)$ is the null sequence. If $j\geq 2$,
then $\textrm{mes}(\gamma,\prec)=(v_{1},\ldots,v_{j-1})$ is a finite sequence
of length $j-1$ such that $v_{1}=\min(\gamma\setminus\gamma_{1})$ and for each
$k\in\\{2,\ldots,j-1\\}$,

$v_{k}=\begin{cases}\min(\\{v_{1},\dots,v_{k-1}\\}\cap(\gamma\setminus\gamma_{k}))&\text{if
}\\{v_{1},\dots,v_{k-1}\\}\cap(\gamma\setminus\gamma_{k})\neq\emptyset,\\\
\min(\gamma\setminus\gamma_{k})&\text{otherwise.}\end{cases}$

Let $M(\gamma,\prec)$ denote the set of vertices appearing in
$\textrm{mes}(\gamma,\prec)$. Define

$d(X,\prec):=\max_{\gamma\in X}|M(\gamma,\prec)|.$

The following result gives us a bound for the collapsibility number of the
complex $X$ using $d(X,\prec)$.

######  Proposition 3.2.

[12, Theorem 6] If $\prec$ is a linear ordering of the maximal simplices of
$X$, then $X$ is $d(X,\prec)$-collapsible.

The proof of [12, Theorem 6] can be modified to show that $\theta_{0}(X)$ is
bounded above by $d(X,\prec)$.

######  Proposition 3.3.

Let $X$ be a simplicial complex, then $\theta_{0}(X)\leq d(X,\prec)$.

###### Proof.

The proof is by induction on number of vertices of $X$. If $X$ is a simplex
then $\theta_{0}(X)=0\leq d(X,\prec)$. If $X$ has at least one non-cone vertex
$v$, then by definition
$\theta_{0}(X)\leq\max\\{\theta_{0}(\mathrm{del}(v,X)),\theta_{0}(\mathrm{lk}(v,X))+1\\}$.

The argument in [12, Theorem 6] shows that

$d(\mathrm{lk}(v,X),\prec)-1\leq d(X,\prec)\text{ and
}d(\mathrm{del}(v,X),\prec)\leq d(X,\prec).$

Hence the proof follows by induction. ∎

For a positive integer $n$, let $[n]$ denotes the ordered set
$\\{1,\ldots,n\\}$. In the rest of the section, we assume that
$V(\mathcal{H})=[n]$ and if $e=\\{v_{1},v_{2},\dots,v_{r}\\}\in
E(\mathcal{H})$ then $v_{1}<v_{2}<\cdots<v_{r}$. Let $<_{L}$ be the
lexicographic order on the $E(\mathcal{H})$ induced by the ordering of
$V(\mathcal{H})$. Note that every facet of $\mathrm{NC}(\mathcal{H})$ is the
complement of an edge of $\mathcal{H}$. We define a linear order $\prec$ on
the set of facets of $\mathrm{NC}(\mathcal{H})$ as follows:
$\gamma\prec\gamma^{\prime}$ if
$\overline{\gamma}<_{L}\overline{\gamma^{\prime}}$. Let
$\gamma_{1}\prec\gamma_{2}\prec\ldots\prec\gamma_{m}$ be all the facets of
$\mathrm{NC}(\mathcal{H})$.

Let $D$ be a cover of $\mathcal{H}$ such that $|D|=\tau(\mathcal{H})$. Without
loss of generality, we assume that $D=\\{1,\ldots,\tau(\mathcal{H})\\}$. For
$A\subseteq[n]$, we let $\overline{A}=[n]\setminus A$.

######  Lemma 3.4.

Let $\gamma,\gamma^{\prime}\in\mathrm{NC}(\mathcal{H})$. If
$\overline{\gamma}\cap D=\overline{\gamma^{\prime}}\cap D$ and the induced
subgraph $\mathcal{H}[\overline{\gamma}\cap D]$ contains an edge, then

$\text{mes}(\gamma,\prec)=\text{mes}(\gamma^{\prime},\prec).$

###### Proof.

Let $k$ be the smallest index such that $\gamma\subseteq\gamma_{k}$. Since
$\mathcal{H}[\overline{\gamma}\cap D]$ contains an edge, the edge
$\overline{\gamma_{k}}\subseteq D$. Observe that for any edge $e^{\prime}\in
E(\mathcal{H}[D]$), and an edge $e\in E(\mathcal{H})$ such that $e\cap
D\neq\emptyset$ and $e\cap\overline{D}\neq\emptyset$, we have
$e^{\prime}<_{L}e$. Therefore, by the definition of $\prec$, it follows that
for every $1\leq i<k$, the $i$th facet $\gamma_{i}$ satisfies
$\overline{\gamma_{i}}\subseteq D$ (since
$\overline{\gamma_{i}}<_{L}\overline{\gamma_{k}}\subseteq D$). Clearly,
$\gamma\cap D=\gamma^{\prime}\cap D$. Hence, we get that

$\overline{\gamma_{i}}\cap\gamma=\overline{\gamma_{i}}\cap\gamma\cap
D=\overline{\gamma_{i}}\cap\gamma^{\prime}\cap
D=\overline{\gamma_{i}}\cap\gamma^{\prime}.$

Therefore, we conclude that if $k$ is the smallest index such that
$\gamma^{\prime}\subseteq\gamma_{k}$ and for every $i\in[k-1]$, then the $i$th
entry of $\textrm{mes}(\gamma^{\prime},{\prec})$ is equal to the $i^{th}$
entry that of $\textrm{mes}(\gamma,{\prec})$. ∎

######  Lemma 3.5.

For any $S\subseteq D$, $N(S)\neq\emptyset$ and

$|N(S)\cap\overline{D}|-|S|\leq|\overline{D}|-|D|.$

###### Proof.

If $S=D$, then result vacuously true. So, assume that
$D\setminus\neq\emptyset$. We prove this by defining an injective map
$\phi:D\setminus S\longrightarrow\overline{D}\setminus(N(S)\cap\overline{D})$.
For any $v\in D$, let $NC(D,v)=\\{e\in E(\mathcal{H}):(D\setminus\\{v\\})\cap
e=\emptyset\\}$. Let $D\setminus S=\\{v_{1},\dots,v_{r}\\}$. Define

$\phi_{1}:\\{v_{1}\\}\longrightarrow\overline{D}\setminus(N(S)\cap\overline{D})$

as $\phi_{1}(v_{1})=\text{min}\big{(}\bigcup\limits_{e\in
NC(D,v_{1})}e\setminus\\{v_{1}\\}\big{)}$. For any $v\in D$ and for any $e\in
NC(D,v)$, it is easy to see that
$e\setminus\\{v\\}\subseteq\overline{D}\setminus(N(S)\cap\overline{D})$.
Therefore, the map $\phi_{1}$ is well defined. For each $2\leq j\leq r$, we
will now construct an injective map

$\phi_{j}:\\{v_{1},\dots,v_{j}\\}\longrightarrow\overline{D}\setminus(N(S)\cap\overline{D})$

using the map $\phi_{j-1}$. We divide the construction of $\phi_{j}$ in two
parts.

Case 1: $\big{(}\bigcup\limits_{e\in
NC(D,v_{j})}e\big{)}\setminus\\{v_{j},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}\neq\emptyset$.

In this case define the map $\phi_{j}$ as follows.

$\phi_{j}(v_{i})=\begin{cases}\phi_{j-1}(v_{i}),&\text{ if }i\leq j-1,\\\
\text{min}\big{(}\big{(}\bigcup\limits_{e\in
NC(D,v_{j})}e\big{)}\setminus\\{v_{j},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}\big{)},&\text{
if }i=j.\end{cases}$

Case 2: $\big{(}\bigcup\limits_{e\in
NC(D,v_{j})}e\big{)}\setminus\\{v_{j},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}=\emptyset$.

In this case choose a $t\in[j]$ such that $\big{(}\bigcup\limits_{e\in
NC(D,v_{t})}e\big{)}\nsubseteq\\{v_{t},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}$.
If there is no such $t\in[j],$ then
$(D\setminus\\{v_{1},\dots,v_{j+1}\\})\cup\\{\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}$
will be a covering set of size smaller than $|D|$, which is a contradiction.
Let $t\in[j]$ such that $\big{(}\bigcup\limits_{e\in
NC(D,v_{t})}e\big{)}\nsubseteq\\{v_{t},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}$.
Clearly $t\neq j$. Now define

$\phi_{j}:\\{v_{1},\dots,v_{j}\\}\longrightarrow\overline{D}\setminus(N(S)\cap\overline{D})$

as follows.

$\phi_{j}(v_{i})=\begin{cases}\phi_{j-1}(v_{i}),&\text{ if }i\neq t,j,\\\
\phi_{j-1}(v_{t}),&\text{ if }i=j,\\\
\text{min}\big{(}\big{(}\bigcup\limits_{e\in
NC(D,v_{j})}e\big{)}\setminus\\{v_{t},\phi_{j-1}(v_{1}),\dots,\phi_{j-1}(v_{j-1})\\}\big{)},&\text{
if }i=t.\end{cases}$

By construction, the map $\phi_{j}$ is injective. Hence
$\phi=\phi_{r}:D\setminus
S\longrightarrow\overline{D}\setminus(N(S)\cap\overline{D})$ is an injective
map. ∎

We now prove the main result of this section. This is done by extending the
idea of [4] to a new setting.

###### Proof of Theorem 3.1.

For a $\sigma\in\mathrm{NC}(\mathcal{H})$, we let
$\Psi_{\sigma}=|N(\overline{\sigma}\cap
D)\cap\overline{\sigma}\cap\overline{D}|$.

Let $\gamma\in\mathrm{NC}(\mathcal{H})$. We first show that for
$v\in\gamma\cap\overline{D}$, if $v\in M(\gamma,\prec)$, then $v$ is a
neighbour of some vertex in $\overline{\gamma}\cap D$. Let $k$ be the smallest
index such that the $k^{th}$ entry of $\textrm{mes}(\gamma,\prec)$ is $v$.
Then $v\in\gamma\setminus\gamma_{k}$, which implies that
$v\in\overline{\gamma_{k}}$. Since $D$ is a cover and
$\overline{\gamma_{k}}\in E(\mathcal{H})$, $\overline{\gamma_{k}}\cap
D\neq\emptyset$. Choose a $w\in\overline{\gamma_{k}}\cap D$. Since $w<v$ and
$v$ is the $k^{th}$ entry of $\textrm{mes}(\gamma,\prec)$, we see that
$w\notin\gamma$ implying that $w\in\overline{\gamma}\cap D$. Furthermore,
since $v,w\in\overline{\gamma_{k}}$, $v$ is a neighbor of $w$.

Therefore,

$\displaystyle|M(\gamma,\prec)|$ $\displaystyle\leq|\gamma\cap
D|+|N(\overline{\gamma}\cap D)\cap(\gamma\cap\overline{D})|$
$\displaystyle=|D|-|\overline{\gamma}\cap D|+|N(\overline{\gamma}\cap
D)\cap\overline{D}|-\Psi_{\gamma}$
$\displaystyle\leq|D|-\tau(\mathcal{H})+|\overline{D}|-\Psi_{\gamma}$
$\displaystyle=|V(\mathcal{H})|-\tau(\mathcal{H})-\Psi_{\gamma},$ (1)

where the last inequality holds by applying Claim 3.5 to the set
$\overline{\sigma}\cap D$.

By Proposition 3.2, it is sufficient to show that $\Psi_{\gamma}\geq 1$.
Suppose that $\Psi_{\gamma}=0$. There exists an edge $e$ such that
$e\subseteq\overline{\gamma}$. If $e\cap\overline{D}\neq\emptyset$, then
$\Psi_{\gamma}=N(\overline{\gamma}\cap
D)\cap\overline{\gamma}\cap\overline{D}\neq\emptyset$. Hence we conclude that
$\mathcal{H}[\overline{\gamma}\cap D]$ has an edge. Let
$\gamma^{\prime}=\gamma\cap D$. Then $\overline{\gamma}\cap
D=\overline{\gamma^{\prime}}\cap D$. By Lemma 3.4,
$\textrm{mes}(\gamma,\prec)=\textrm{mes}(\gamma^{\prime},\prec)$ and therefore
$M(\gamma,\prec)=M(\gamma^{\prime},\prec)$. Observe that
$\Psi_{\gamma^{\prime}}\geq 1$. Thus, by replacing $\gamma$ by
$\gamma^{\prime}$ and using Section 3, we conclude that
$|M(\gamma,\prec)|\leq|V(\mathcal{H})|-\tau(\mathcal{H})-1$. Result follows
from Proposition 3.3. ∎

A set $B\subseteq V(\mathcal{H})$ is called a dominating set of $\mathcal{H}$
if every vertex in $V(\mathcal{H})\setminus B$ is a neighbour of at least one
vertex from $B$. The domination number of $\mathcal{H}$ is

$\gamma(\mathcal{H})=\min\\{|B|:B\text{ is a dominating set of
}\mathcal{H}\\}.$

It is easy to observe that $\gamma(\mathcal{H})\leq\tau(\mathcal{H})$ for
every hypregraph $\mathcal{H}$. Therefore the following result is an immediate
corollary of Theorem 3.1 and Theorem 2.5.

######  Corollary 3.6.

For any hypergraph $\mathcal{H}$,

$\mathcal{C}(\mathrm{NC}(\mathcal{H}))\leq\theta(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\gamma(\mathcal{H})-1.$

We now compare Theorem 3.1 with the results of Kim and Kim [11], where they
established upper bounds on the Leray number of $\mathrm{NC}(\mathcal{H})$
with various domination parameters of the hypergraphs $\mathcal{H}$. To do
this comparison, we first recall the required terminology from [11].

Let $\mathcal{H}$ be a hypergraph. Let $v\in V(\mathcal{H})$ and $B$ be a
subset of $V(\mathcal{H})$. Then $B$ strongly totally dominates $v$ if there
exists $B^{\prime}\subseteq B\setminus\\{v\\}$ such that
$B^{\prime}\cup\\{v\\}\in E(\mathcal{H})$.

Let $W$ be a subset of $V(\mathcal{H})$. If $B\subseteq V$ strongly dominates
every vertex in $W$, then $B$ is said to be strongly dominates $W$.

The strong total domination number of $A$ in $\mathcal{H}$ is defined as

$\gamma(\mathcal{H};W):=\min\\{|B|:B\subseteq V(\mathcal{H}),~{}B\text{
strongly dominates }W\\}.$

######  Definition 3.7.

The strong total domination number $\tilde{\gamma}(\mathcal{H})$ of
$\mathcal{H}$ is the strong total domination number of $V(\mathcal{H})$, i.e.,
$\tilde{\gamma}(\mathcal{H})=\gamma(\mathcal{H};V(\mathcal{H}))$.

A set $\mathcal{I}\subseteq V(\mathcal{H})$ is said to be strongly independent
in $\mathcal{H}$ if it is independent and every edge of $\mathcal{H}$ contains
at most one vertex of $\mathcal{I}$.

######  Definition 3.8.

The strong independence domination number of $\mathcal{H}$ is the integer

$\gamma_{si}(\mathcal{H}):=\max\\{\gamma(\mathcal{H};\mathcal{I}):\mathcal{I}\text{
is a strongly independent set of }\mathcal{H}\\}.$

######  Definition 3.9.

The edgewise-domination number of $\mathcal{H}$ is the minimum number of edges
whose union strongly dominates the $V(\mathcal{H})$, i.e.

$\gamma_{E}(\mathcal{H}):=\min\\{|\mathcal{F}|:\mathcal{F}\subseteq
E(\mathcal{H}),\bigcup_{e\in\mathcal{F}}e\text{ strongly dominates
}V(\mathcal{H})\\}.$

######  Theorem 3.10.

[11, Theorem 1.6] Let $\mathcal{H}$ be a hypergraph with no isolated vertices.
Then

  1. (i)

If $|e|\leq 3$ for every $e\in E(\mathcal{H})$, then
$L(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\left\lceil\frac{\tilde{\gamma}(\mathcal{H})}{2}\right\rceil-1$.

  2. (ii)

If $|e|\leq 2$ for every $e\in E(\mathcal{H})$, then
$L(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\gamma_{si}(\mathcal{H})-1$.

  3. (iii)

$\mathcal{L}(\mathrm{NC}(\mathcal{H}))\leq|V(\mathcal{H})|-\gamma_{E}(\mathcal{H})-1$.

######  Example 3.11.

Let $n\geq 5$ be a positive integer. Let $\mathcal{H}=([n],\binom{n}{3})$. For
any set $A\subseteq[n]$ such that $|A|\leq n-3$, there exists and edge
$e\subseteq[n]\setminus A$ and therefore $\tau(\mathcal{H})\geq n-2$. Since
$[n-2]\cap e\neq\emptyset$ for all $e\in E(\mathcal{H})$, we conclude that
$\tau(\mathcal{H})=n-2$. Clearly any set of cardinality $3$ strongly dominates
$[n]$. Hence $\gamma_{E}(\mathcal{H})=1$. Since any set of cardinality $1$ or
$2$ can not strongly dominate $[n]$, we see that
$\tilde{\gamma}(\mathcal{H})=3$. Observe that any strongly independent set is
of cardinality $1$ and therefore we conclude that
$\gamma_{si}(\mathcal{H})=2$.

######  Remark 3.12.

Note that for the above class of examples,
$\tau(\mathcal{H})>\max\\{\tilde{\gamma}(\mathcal{H}),\gamma_{si}(\mathcal{H}),\gamma_{E}(\mathcal{H})\\}$.
Wegner [16, Lemma 3] proved that $\mathcal{L}(X)\leq\mathcal{C}(X)$. Thus, for
the class of hypergraphs in Example 3.11, we obtain a much tighter bound on
$\mathcal{L}(X)$ from Theorem 3.1 as compared to those obtained in [11].

##  4 Future directions

In Example 2.4, we gave a simplicial complex $X$ such that
$\theta_{k}(X)<\theta_{k-1}(X)$ for $k=1$. However, we are unable to find
examples for general $k$. Therefore, we pose the problem here.

######  Question 4.1.

Given a $k\geq 2$, does there exist a simplicial complex $X$ such that
$\theta_{k}(X)<\theta_{k-1}(X)$?

In Theorem 2.9, we proved that $\mathcal{C}(X)=\theta_{k}(X)$, if $X$ is
$k$-vertex decomposable. It would be interesting to find classes of simplicial
complexes for which $\theta_{k}$ is equal to the collapsibility number.

######  Question 4.2.

Classify simplicial complexes $X$ for which $\mathcal{C}(X)=\theta_{k}(X)$.

For complexes $X$ and $Y,$ we know that the collapsibility number of join,
$\mathcal{C}(X\ast Y)=\mathcal{C}(X)+\mathcal{C}(Y)$ ([10, Proposition 4.2.3]
and the discussion below it) and $\theta_{0}(X\ast
Y)=\theta_{0}(X)+\theta_{0}(Y)$ ([2, Theorem 11]). This raises the following
natural question.

######  Question 4.3.

For any $k\geq 1$ and the simplicial complexes $X$ and $Y$,

$\theta_{k}(X\ast Y)=\theta_{k}(X)+\theta_{k}(Y).$

## Acknowledgements

We thank the Department of Mathematics, Indian Institute of Technology Bombay,
for hosting us, where the major part of this work was done. The third author
is partially supported by the Start-up Research Grant SRG/2022/000314 from
SERB, DST, India.

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