scholarly journals Matching Complexes of Small Grids

10.37236/8480 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Takahiro Matsushita
Keyword(s):  
Simplicial Complex ◽  
Line Graph ◽  
Simple Graph ◽  
Grid Graph ◽  
Matching Complex ◽  
Independence Complex

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$.  Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.

scholarly journals Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph

10.37236/5660 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Arrigo Bonisoli ◽  
Simona Bonvicini
Keyword(s):  
Perfect Matching ◽  
Line Graph ◽  
Simple Graph ◽  
Connected Components ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Cubic Graph ◽  
Cycle Decomposition ◽  
Odd Degree ◽  
Necessary And Sufficient

Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.

scholarly journals Colouring Squares of Claw-free Graphs

2019 ◽  
Vol 71 (1) ◽  
pp. 113-129 ◽  
Author(s):  
Rémi de Joannis de Verclos ◽  
Ross J. Kang ◽  
Lucas Pastor
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Clique Number ◽  
Simple Graph ◽  
General Question ◽  
Free Graph ◽  
Original Question

AbstractIs there some absolute $\unicode[STIX]{x1D700}>0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\unicode[STIX]{x1D712}(G^{2})\leqslant (2-\unicode[STIX]{x1D700})\unicode[STIX]{x1D714}(G)^{2}$, where $\unicode[STIX]{x1D714}(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case where $G$ is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.

scholarly journals Collapsibility of Non-Cover Complexes of Graphs

10.37236/8684 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ilkyoo Choi ◽  
Jinha Kim ◽  
Boram Park
Keyword(s):  
Simplicial Complex ◽  
Domination Number ◽  
Independent Set ◽  
Chordal Graphs ◽  
Alexander Dual ◽  
Vertex Set ◽  
Independence Complex ◽  
Independence Domination Number

Let $G$ be a graph on the vertex set $V$. A vertex subset $W \subseteq V$ is a cover of $G$ if $V \setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex on $V$ whose faces are non-covers of $G$. Then the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i\gamma(G)-1)$-collapsible where $i\gamma(G)$ denotes the independence domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs.

scholarly journals 3-Total Edge Product Cordial Labeling for Stellation of Square Grid Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Rizwan Ullah ◽  
Gul Rahmat ◽  
Muhammad Numan ◽  
Kraidi Anoh Yannick ◽  
Adnan Aslam
Keyword(s):  
Simple Graph ◽  
Grid Graph ◽  
Square Grid ◽  
Vertex Set ◽  
Edge Labeling

Let G be a simple graph with vertex set V G and edge set E G . An edge labeling δ : E G ⟶ 0,1 , … , p − 1 , where p is an integer, 1 ≤ p ≤ E G , induces a vertex labeling δ ∗ : V H ⟶ 0,1 , … , p − 1 defined by δ ∗ v = δ e 1 δ e 2 ⋅ δ e n mod p , where e 1 , e 2 , … , e n are edges incident to v . The labeling δ is said to be p -total edge product cordial (TEPC) labeling of G if e δ i + v δ ∗ i − e δ j + v δ ∗ j ≤ 1 for every i , j , 0 ≤ i ≤ j ≤ p − 1 , where e δ i and v δ ∗ i are numbers of edges and vertices labeled with integer i , respectively. In this paper, we have proved that the stellation of square grid graph admits a 3-total edge product cordial labeling.

scholarly journals Matching Complexes of $3 \times n$ Grid Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Shuchita Goyal ◽  
Samir Shukla ◽  
Anurag Singh
Keyword(s):  
Simplicial Complex ◽  
Homotopy Equivalent ◽  
Grid Graphs ◽  
Comprehensive List ◽  
Matching Complex

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed  that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.  

scholarly journals Matching and Independence Complexes Related to Small Grids

10.37236/6212 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Benjamin Braun ◽  
Wesley K. Hough
Keyword(s):  
Euler Characteristic ◽  
Grid Graph ◽  
Homology Groups ◽  
Matching Complex

The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.

scholarly journals Some improved bounds on two energy-like invariants of some derived graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 883-893
Author(s):  
Shu-Yu Cui ◽  
Gui-Xian Tian
Keyword(s):  
Theoretical Analysis ◽  
Lower Bounds ◽  
Regular Graph ◽  
Line Graph ◽  
Simple Graph ◽  
Square Root ◽  
Laplacian Eigenvalues ◽  
Graph Theoretical Analysis ◽  
Laplacian Energy ◽  
Signless Laplacian

Abstract Given a simple graph G, its Laplacian-energy-like invariant LEL(G) and incidence energy IE(G) are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. This paper obtains some improved bounds on LEL and IE of the 𝓡-graph and 𝓠-graph for a regular graph. Theoretical analysis indicates that these results improve some known results. In addition, some new lower bounds on LEL and IE of the line graph of a semiregular graph are also given.

scholarly journals On the k-Buchsbaum property of powers of Stanley–Reisner ideals

2014 ◽  
Vol 213 ◽  
pp. 127-140 ◽  
Author(s):  
Nguyên Công Minh ◽  
Yukio Nakamura
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Residue Class ◽  
Simple Graph ◽  
Edge Ideal ◽  
Vertex Set

AbstractLet S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

scholarly journals A Construction of Sequentially Cohen–Macaulay Graphs

2021 ◽  
Vol 28 (03) ◽  
pp. 399-414
Author(s):  
Aming Liu ◽  
Tongsuo Wu
Keyword(s):  
Simplicial Complex ◽  
Betti Numbers ◽  
Simple Graph ◽  
Ideal Formula ◽  
Vertex Decomposable ◽  
Linear Quotients ◽  
Facet Ideal

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

scholarly journals 3-torsion in the Homology of Complexes of Graphs of Bounded Degree

2013 ◽  
Vol 65 (4) ◽  
pp. 843-862
Author(s):  
Jakob Jonsson
Keyword(s):  
Boundary Element ◽  
Simplicial Complex ◽  
Homology Class ◽  
Chain Complex ◽  
Semigroup Algebras ◽  
Bounded Degree ◽  
Matching Complex ◽  
Image Position

AbstractFor δ ≥ 1 and n ≥ 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When δ = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n − 4)/3 ≤ d ≤ (n − 6)/2. This paper establishes similar bounds for δ ≥ 2. Specifically, there is 3-torsion in degree d wheneverThe procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3zis a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

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