Karen L. Collins, Megan E. Heenehan, Jessica McDonald
Let $G,H$ be graphs and $G*H$ represent a particular graph product of $G$ and $H$. We define $im(G)$ to be the largest $t$ such that $G$ has a $K_t$-immersion and ask: given $im(G)=t$ and $im(H)=r$, how large is $im(G*H)$? Best possible lower bounds are provided when $*$ is the Cartesian or lexicographic product, and a conjecture is offered for each of the direct and strong products, along with some partial results.
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Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t + 4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no stable set of size three contains $K_{2\lfloor n/5 \rfloor}$ as an immersion.
Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$.
How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $2^{t+\log^2 t}n$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$.…
A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $ϕ: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $ϕ(u)$ and $ϕ(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel, and independently by Abu-Khzam and Langston, states that every graph $G$ contains $K_{χ(G)}$ as an immersion. We prove that for any constant $\varepsilon>0$ and integers $s,t\ge2$, there exists $d_0=d_0(\varepsilon,s,t)$ such that…
We prove that if $L(G)$ immerses $K_t$ then $L(mG)$ immerses $K_{mt}$, where $mG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel edge of multiplicity $m$. This implies that when $G$ is a simple graph, $L(mG)$ satisfies a conjecture of Abu-Khzam and Langston. We also show that when $G$ is a line graph, $G$ has a $K_t$-immersion iff $G$ has a $K_t$-minor whenever $t\leq 4$, but this equivalence fails in both directions when $t \geq 5$.
Clique immersions and independence number
本文研究图的团浸入与独立数的关系，针对所有独立数不小于3的情形，改进了 Gauthier、Le 和 Wollan 关于团浸入规模的已有下界，给出了更优的顶点数保证。
Counterexamples to Clique Immersion Conjecture for Direct Products
针对Collins等人提出的关于图直积的团浸入猜想，构造了一个无限连通二分图族作为反例。
Coloring Graphs With No Totally Odd Clique Immersion
证明无完全奇K_t浸入的图是O(t)-可着色的，并给出固定参数可解算法找到分解。
Clique immersions in graphs of independence number two with certain forbidden subgraphs
本文研究独立数为2的图中的Lescure-Meyniel猜想，证明无特定长度洞的图及无某些小禁用子图的图均满足该浸入猜想。
Every properly colored graph with $χ(G)=k$ colors has edge-disjoint Kempe "backbones", Kempe chains anchored by color-critical vertices for each pair of colors. Certain color permutations arrange these backbones into a clique-like structure, a strengthening of strong immersions of complete graphs. This strengthened immersion is suggested as a template for identifying the disjoint subgraphs comprising Hadwiger's conjectured $K_k$ minor present in $k$-chromatic graphs.
We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall.
We consider the problem of how much edge connectivity is necessary to force a graph G to contain a fixed graph H as an immersion. We show that if the maximum degree in H is D, then all the examples of D-edge connected graphs which do not contain H as a weak immersion must have a tree-like decomposition called a tree-cut decomposition of bounded width. If we consider strong immersions, then it is easy to see that there are arbitrarily highly edge connected graphs which do not contain a fixed clique K_t as a strong immersion. We give a structure theorem which roughly characterizes those highly e…
Topological cliques in sparse expanders
本文研究稀疏扩展图中团浸入与团细分的存在性，给出三类充分条件下大团浸入或细分的渐近构造，并部分回答了 Dvořák 和 Yepremyan 关于最小度与团浸入的问题。
Near-Optimal Algorithms for Maximal Clique Enumeration in Structurally Sparse Graphs
针对排除团子式和排除团浸入的图，提出两种最大团枚举算法，分别将时间复杂度优化为n·4^(2t/5+o(t))和n·3^(t/3+o(t))，并证明其指数基为渐近最优。
Immersions of large cliques in graphs with independence number 2 and bounded maximum degree
本文研究独立数为2的图中大团浸入问题，证明在最大度及团覆盖数的相应界值下，图必含 $K_{\chi(G)}$ 和 $K_{\lceil n/2\rceil}$ 的浸入，从而验证有界最大度情形下的 Vergara 猜想。
Immersions and Albertson's conjecture
本文证明了Lescure-Meyniel关于弱浸入的猜想在顶点数至多$(1.64-o(1))k$的图上成立，并由此推进了Albertson猜想，即色数为k的图其交叉数至少为$cr(K_k)$。
An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and $f(v)$. The immersion is strong if the paths $P_{uv}$ are internally disjoint from $f(V(H))$. It is proved that for every positive integer $t$, every simple graph of minimum degree at least $200t$ contains a strong immersion of the complete graph $K_t$. For dense graphs one can say even more. If the graph has order $n$ and has $2cn^2$ edges, then there is a…
Totally odd immersions of complete graphs in graph products
研究完全图在图积中的完全奇浸入问题，并证明无法从四种标准图积中获得奇Hadwiger猜想的浸入版本的最小反例。
An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. It was recently shown by Gravier, Moncel and Semri that the minimum cardinality of an identifying code for the Cartesian product of two cliques of the same order n is the floor of 3n/2. We consider identifying codes of the direct product of two cliques. In particular, we answer a question of Klavzar and determine the minimum cardinality of an identifying code for the direct product of any two cliques.
An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of the Cartesian product of two cliques with the same size. In this note, we generalize this result to the Cartesian product of any two nontrivial cliques.
For a graph $X$ without isolated vertices and without isolated edges, a product-irregular labelling $ω:E(X)\rightarrow \{1,2,\ldots,s\}$, first defined by Anholcer in 2009, is a labelling of the edges of $X$ such that for any two distinct vertices $u$ and $v$ of $X$ the product of labels of the edges incident with $u$ is different from the product of labels of the edges incident with $v$. The minimal $s$ for which there exist a product irregular labeling is called the product irregularity strength of $X$ and is denoted by $ps(X)$. Clique cover number of a graph is the minimum number of cliques…
In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of arbitrary graphs G and H based on their respective clique numbers.
Let $G$ be a connected graph and $W=\{ w_1, w_2, \ldots, w_k \} \subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1), d(v,w_2), \ldots, d(v,w_k))$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension and is denoted by $dim(G)$. In this paper, we study the metric dimension of…
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\mathcal{C} = \{C_1,C_2, \cdots, C_k\}$ be a clique cover of $G$ and $U\subseteq V(H)$. We consider clique cover product which denoted by $G^\mathcal{C} \star H^U$ and obtained from $G$ as follows: for each clique $C_i \in \mathcal{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the domination polynomial of clique cov…
The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula invol…
In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in "A general 2-part Erdős-Ko-Rado theorem" by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on $f(k,N)$, and we solve the problem up to a constant deviation depending only on $k$, and find the exact value for $f(2,N)$ if $N$ is large enough. We also comp…
The product dimension of a graph $G$ is the minimum possible number of proper vertex colorings of $G$ so that for every pair $u,v$ of non-adjacent vertices there is at least one coloring in which $u$ and $v$ have the same color. What is the product dimension $Q(s,r)$ of the vertex disjoint union of $r$ cliques, each of size $s$? Lovász, Nešetřil and Pultr proved in 1980 that for $s=2$ it is $(1+o(1)) \log_2 r$ and raised the problem of estimating this function for larger values of $s$. We show that for every fixed $s$, the answer is still $(1+o(1)) \log_2 r$ where the $o(1)$ term tends to $0$ …
We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.
A Proof of Albertson's Conjecture
本文解决了阿尔伯森猜想中r=25、26的三类剩余情形，结合已发表结果及附录对19≤r≤24的重证，证明该猜想对所有r≤26成立，还得出色数为27且交叉数小于cr(K_27)的图的相关子图性质。
The Domination and Secure Domination Numbers of Direct Product of Cliques with Paths and Cycles
该研究确定了完全图与路径/圈直积图的五类控制参数精确值，证明了部分参数间的相等关系，并给出反例纠正了现有文献的错误结论。
Hereditary Graph Product Structure and $\cal H$-clique-width
本文提出H-clique宽度作为图的新型结构度量，用于继承传统图积结构。通过路径类等图类，H-clique宽度定义为图G可表示为强积图的诱导子图的最小整数t。证明了平面图积结构定理等可由诱导子图包含关系表述。
Algebraic study on rooted products of graphs and multi-clique corona graphs
本文研究图的根积的代数性质，定义多团冠图并证明其顶点可分解性，同时给出边理想的相关代数不变量公式。
A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is denoted by $lcc(G)$. Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges. In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed. Moreover, the local clique cover number of claw-f…
The distinguishing number of a graph $G$, denoted $D(G)$, is the minimum number of colors needed to produce a coloring of the vertices of $G$ so that every nontrivial isomorphism interchanges vertices of different colors. A list assignment $L$ on a graph $G$ is a function that assigns each vertex of $G$ a set of colors. An $L$-coloring of $G$ is a coloring in which each vertex is colored with a color from $L(v)$. The list distinguishing number of $G$, denoted $D_{\ell}(G)$ is the minimum $k$ such that every list assignment $L$ that assigns a list of size at least $k$ to every vertex permits a …
Clique structure and other network properties of the tensor product of Erdős-Rényi graphs
该研究分析Erdős-Rényi图张量积的团数量与最大团大小，研究扩展聚类系数及与局部效率的关系，分析孤立顶点数，并探讨随机图模积与最大公共子图问题的关联。
A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindr…
Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots, C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\mathscr{C}}\star H^U$, as follows: for each clique $C_i\in \mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\mathscr{C}}\star H^U$ $$I(G^{\mathscr{C}}\star H^U;x)=I^q(H;x)I(G;\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and …
Weak saturation of tensor product of cliques
研究张量积 clique 的弱饱和度，推广了 Moshkovitz 和 Shapira 的结果，并探讨了着色超图的弱饱和度。
We study the following question raised by Erdős and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of size $\log n$? We construct examples showing that $m$ is at most $2^{2^{(\log\log n)^{1/2+o(1)}}}$ obtaining a twofold sub-polynomial improvement over the upper bound of about $\sqrt{n}$ coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets …
It is known that the number of directions formed by a Cartesian product $A \times B \subset AG(2,p)$ is at least $|A||B| - \min\{|A|,|B|\} + 2$, provided $p$ is prime and $|A||B|<p$. This implies the best known upper bound on the clique number of the Paley graph over $\mathbb{F}_p$. In this paper, we extend this result to $AG(2,q)$, where $q$ is a prime power. We also give improved upper bounds on the clique number of generalized Paley graphs over $\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivial upper bound $\sqrt{q}$ to $0.769\sqrt{q}+1$. In general, as an applic…
Given an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the seq…
We state and prove some counting formulas relating to cliques in the distant graphs of projective lines over finite rings. As a preliminary to this, we prove a decomposition theorem for the graphs in terms of the direct-product decomposition of their rings.
In this paper we enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. The weak modular product differs from the direct product by also encoding non-adjacencies of the factor graphs in its edges. This work is motivated by the following: a 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Furthermore, a straightforward corollary of Kozen's theorem and Lovász's sandwich theorem is if the weak modular product betwee…
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log\log(n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded …
Monophonic position sets of Cartesian and lexicographic products of graphs
本文针对图的笛卡尔积与字典积，研究将一般位置问题中最短路径替换为诱导路径的单音位置问题，推导两类积图单音位置数的相关结论与计算公式。
A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In…
It was recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $ω\geq \frac 23(Δ+1)$ unless the graph is the strong product of $K_{ω/2}$ and an odd hole. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.
Cliques and independent subgroups of the Birkhoff polytope graph
研究Birkhoff多面体图的团和独立子群结构，证明了关于Sym(n)中3循环排列子集的最大大小的Erdős-Ko-Rado型定理。
This paper is an endeavor to discuss some properties of zero-divisor graphs of the ring $\mathbb{Z}_n$, the ring of integers modulo $n$. The zero divisor graph of a commutative ring $R$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$, where two distinct vertices are adjacent if their product is zero. The zero divisor graph of $R$ is denoted by $Γ(R)$. We discussed $Γ(\mathbb{Z}_n)$'s by the attributes of completeness, k-partite structure, complete k-partite structure, regularity, chordality, $γ- β$ perfectness, simplicial vertices. The clique number for arbitrary $Γ…
Strongly regular graphs from hyperbolic quadrics and their maximal cliques
本文研究了从双曲二次曲面构造的强正则图，证明其参数及谱性质，并分类了q=2时的极大团。
Hilton-Milner Theorem for the $r$-independent sets in a union of cliques
本文提出了一种针对多个完全子图并集中的 $r$-独立集的希尔顿-米勒定理，扩展了之前对深度二爪图的研究，验证了Holroyd--Talbot猜想。
Nordhaus-Gaddum inequalities for the number of cliques in a graph
本文研究了图中团数量的Nordhaus-Gaddum不等式，回顾了Ramsey多重性等相关问题的历史，并探讨了固定尺寸版本及多色版本下团数量之和与乘积的界。
The independence and clique cover numbers of the squarefree graph
本文确定了 {1,...,n} 中任意两数乘积均非无平方因子数的最大子集，证明其最大规模由奇无平方因子数的补集达到，从而解决了 Erdős 和 Sárközy 在 1992 年提出的问题。
Let $Γ$ be the graph on the roots of the $E_8$ root system, where any two distinct vertices $e$ and $f$ are connected by an edge with color equal to the inner product of $e$ and $f$. For any set $c$ of colors, let $Γ_c$ be the subgraph of $Γ$ consisting of all the $240$ vertices, and all the edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs, of $Γ$ that are either monochromatic, or of size at most $3$, or a maximal clique in $Γ_c$ for some color set $c$, or whose vertices are the vertices of a face of the $E_8$ root polytope. We prove that, apart from two exceptions,…
Clique number of xor-powers of Kneser graphs
该研究围绕Kneser图异或幂的团数展开，确定了相关参数的量级，推导了特定情形下团数的上下界，并给出团数量级的范围与相关猜想。
Dual Systolic Graphs
本文定义了对偶收缩图并证明其精确等周不等式，同时提出弱对偶收缩性概念，通过团乘积实现高效构造，并揭示其小集合扩展能力与阈值秩的联系。
Odd Hadwiger number and graph products
本文研究了两个图积的奇Hadwiger数大小，针对四种标准图积给出最优下界。
Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $\mathbb{Z}_{h}^{n}$ be the $n$-dimensional row vector space over $\mathbb{Z}_{h}$. A generalized Grassmann graph for $\mathbb{Z}_{h}^n$, denoted by $G_r(m,n,\mathbb{Z}_{h})$ ($G_r$ for short), has all $m$-subspaces of $\mathbb{Z}_{h}^n$ as its vertices, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\leq r\leq m+1\leq n$. In this paper, we determine the clique number and geometric structu…
Exact values and improved bounds on the clique number of cyclotomic graphs
本文研究了循环图的团数，给出了改进的上界，并首次确定了非平方阶广义Paley图的团数。
Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs
本文研究了超几何均匀盘图的乘积结构与树宽，证明了具有常数团数的HUDGs不具有乘积结构，且当半径随顶点数增长时，HUDGs具有乘积结构。
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