Nati Linial, Avraham Morgenstern
For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar phenomenon is established as well for tournaments with k=l=4.
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In this article we present the idea of clique ceiling numbers of the vertices of a given graph that has a universal vertex. We follow up with a polynomial-time algorithm to compute an upper bound for the clique number of such a graph using clique ceiling numbers. We compare this algorithm with some upper bound formulas for the clique number.
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…
Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $δ\ge 6$
本文完整刻画了最小度至少为6的局部循环图中团收敛与团发散的条件，证明团发散等价于泛三角覆盖中存在任意大的三角形状图子结构。
Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the r…
We prove that the clique graph operator $k$ is divergent on a locally cyclic graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $δ(G)=6$ if and only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the maximal complete subgraphs of $G$ as vertices, and the edges are given by non-empty intersections. If all iterated clique graphs of $G$ are pairwise non-isomorphic, the graph $G$ is $k$-divergent; otherwise, it is $k$-convergent. To prove our claim, we explicitly construct the iterated clique graphs of those infinite locally cyclic graphs with $δ\geq6$ which induce simply c…
Local Clique Covers and Chromatic Number
本文证明了局部团覆盖数与色数之和不超过顶点数加一的猜想，并给出了达到等号时图的结构性质及诱导匹配的改进界。
We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5ω(G)}{4}\rceil$ and $\lceil\frac{Δ(G) + ω(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or …
We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a "$c$-closed" graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c…
Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such that for any balanced $r$-colouring of the vertices of $G$ one can find an independent set which meets at least $m$ colour classes in a set of size $|V|$. Answering a conjecture of S. Thomassé, we express the exact value of $r(H_n,m)$ (using Ramsey-numbers for finite digraphs), where $H_n$ is Henson's countable universal homogeneous $K_n$-free graph. In turn…
Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i.e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and Δ, we denote by T(n,Δ) the class of all n-vertex trees with maximum degree at most Δ. In this work…
The celebrated Erdős-Hajnal Conjecture says that in any proper hereditary class of finite graphs we are guaranteed to have a clique or anti-clique of size $n^c$, which is a much better bound than the logarithmic size that is provided by Ramsey's Theorem in general. On the other hand, in uncountable cardinalities, the model-theoretic property of stability guarantees a uniform set much larger than the bound provided by the Erdős-Rado Theorem in general. Even though the consequences of stability in the finite have been much studied in the literature, the countable setting seems a priori quite dif…
Faithful universal graphs for minor-closed classes
研究minor-闭类的忠实通用图，证明K_t-minor-无且含所有n顶点平面图的图需2^{Ω(n)}顶点，构造多项式大小的K_4-minor-无图（含所有n顶点树）和K_7-minor-无图（含所有n顶点K_4-minor-无图），回答相关问题并推广研究各类图的通用图阶。
Bell coloring graphs: realizability and reconstruction
该研究探讨了贝尔着色图的结构特性，证明了树和环可以作为贝尔着色图出现，并揭示了某些图类无法成为贝尔着色图的条件，同时给出了关于贝尔着色图作为不变量的重建结果。
The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry
研究环多项式L_G，证明其包含Tutte多项式并统一多个图不变量，提出与U多项式区分能力等价的猜想，并解决Merino和Noble的开放问题。
Hypergraph Universality and Erdos-Posa Failure for Chromatically Rich Odd Cycles
本文证明对任意k≥3，任意有限丛可由k-坏奇圈的极小终端迹实现，保持打包-覆盖数，导致有限拥塞和分数Erdős-Pósa性质失效，与k=2情形形成对比。
A box in Euclidean $k$-space is the Cartesian product $I_1\times I_2\times \cdots \times I_k$, where $I_j$ is a closed interval on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space. Mycielski introduced an interesting graph operation that extends a graph $G$ to a new graph $M(G)$, called the Mycielski graph of $G$. In this paper, we observe behavior of the boxicity of Mycielski graphs. The inequality $\text{box}(M(G))\geq \text{box}(…
Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth
本文研究关于无奇洞图树宽障碍的猜想，证明了t=4和H为锥森林的特殊情况，强化了猜想并推进到H为晶体的情况，同时指出其强化版本已被反驳。
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