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Register a new userOn the minimal degree condition of graphs implying some properties of subgraphs
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ErdH{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let $delta_r$ be the least number so that any graph $G$ on $n$ vertices with minimum degree $delta_rn$ has the property $P_{r-1}(G)=K_rf(G),$ where $P_{r-1}(G)$ is the largest number of edges in an $(r-1)$-partite subgraph and $K_rf(G)$ is the largest number of edges in a $K_r$-free subgraph. We show that $frac{3r-4}{3r-1}<delta_rlefrac{4(3r-7)(r-1)+1}{4(r-2)(3r-4)}$ when $rge4.$ In particular, $delta_4le 0.9415.$
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We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, ErdH{o}s and Lov{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.
A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,ldots,H_q)$, denoted by $Gto_q(H_1,ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $iin[q]$. Let $s_q(H_1,ldots,H_q)$ denote the smallest minimum degree of $G$ over all graphs $G$ that are minimal $q$-Ramsey for $(H_1,ldots,H_q)$ (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, ErdH{o}s and Lovasz, who determined its value precisely for a pair of cliques. Over the past two decades the parameter $s_q$ has been studied by several groups of authors, the main focus being on the symmetric case, where $H_icong H$ for all $iin [q]$. The asymmetric case, in contrast, has received much less attention. In this paper, we make progress in this direction, studying asymmetric tuples consisting of cliques, cycles and trees. We determine $s_2(H_1,H_2)$ when $(H_1,H_2)$ is a pair of one clique and one tree, a pair of one clique and one cycle, and when it is a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on $s_q(C_ell,ldots,C_ell,K_t,ldots,K_t)$ in terms of the size of the cliques $t$, the number of cycles, and the number of cliques. Our bounds are tight up to logarithmic factors when two of the three parameters are fixed.
Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large~$k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $min(D)geq 3$ and $3,4in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.
Given any graph $H$, a graph $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges of $G$ with $q$ colors yields a monochromatic subgraph isomorphic to $H$. Further, such a graph $G$ is said to be minimal $q$-Ramsey for $H$ if additionally no proper subgraph $G$ of $G$ is $q$-Ramsey for $H$. In 1976, Burr, ErdH{o}s, and Lovasz initiated the study of the parameter $s_q(H)$, defined as the smallest minimum degree among all minimal $q$-Ramsey graphs for $H$. In this paper, we consider the problem of determining how many vertices of degree $s_q(H)$ a minimal $q$-Ramsey graph for $H$ can contain. Specifically, we seek to identify graphs for which a minimal $q$-Ramsey graph can contain arbitrarily many such vertices. We call a graph satisfying this property $s_q$-abundant. Among other results, we prove that every cycle is $s_q$-abundant for any integer $qgeq 2$. We also discuss the cases when $H$ is a clique or a clique with a pendant edge, extending previous results of Burr et al. and Fox et al. To prove our results and construct suitable minimal Ramsey graphs, we develop certain new gadget graphs, called pattern gadgets, which generalize and extend earlier constructions that have proven useful in the study of minimal Ramsey graphs. These new gadgets might be of independent interest.
This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k$ and $Delta$, every graph with maximum degree at most $Delta$ and sufficiently large treewidth contains either a subdivision of the $(ktimes k)$-wall or the line graph of a subdivision of the $(ktimes k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For $tgeq 2$, a $t$-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least $t$. A $t$-pyramid is a graph consisting of a vertex $v$, a triangle $B$ disjoint from $v$ and three paths starting at $v$ and disjoint otherwise, each joining $v$ to a vertex of $B$, and each of length at least $t$. We prove that for all $k,t$ and $Delta$, every graph with maximum degree at most $Delta$ and sufficiently large treewidth contains either a $t$-theta, or a $t$-pyramid, or the line graph of a subdivision of the $(ktimes k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a $t$-theta for some $tgeq 2$). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every $Delta$ and subcubic subdivided caterpillar $T$, every graph with maximum degree at most $Delta$ and sufficiently large treewidth contains either a subdivision of $T$ or the line graph of a subdivision of $T$ as an induced subgraph.
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