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تسجيل مستخدم جديدMinimal Ramsey graphs for cyclicity
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We study graphs with the property that every edge-colouring admits a monochromatic cycle (the length of which may depend freely on the colouring) and describe those graphs that are minimal with this property. We show that every member in this class reduces recursively to one of the base graphs $K_5-e$ or $K_4vee K_4$ (two copies of $K_4$ identified at an edge), which implies that an arbitrary $n$-vertex graph with $e(G)geq 2n-1$ must contain one of those as a minor. We also describe three explicit constructions governing the reverse process. As an application we are able to establish Ramsey infiniteness for each of the three possible chromatic subclasses $chi=2, 3, 4$, the unboundedness of maximum degree within the class as well as Ramsey separability of the family of cycles of length $leq l$ from any of its proper subfamilies.
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Given graphs $H_1, dots, H_t$, a graph $G$ is $(H_1, dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $iin{1, dots, t}$, but any proper subgraph of $G $ does not possess this pr
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For an integer $qge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called
$q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Nev{s}etv{r}il and Rodl from 1977 we prove that, for $qge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences. (1) For $2le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$. (2) For every $qge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection ${{cal M}_q(H) : H text{ is 3-connected or } K_3}$ forms an antichain with respect to the subset relation, where ${cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$. We also address the question which pairs of graphs satisfy ${cal M}_q(H_1)={cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $qge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nev{s}etv{r}il and Rodl and by Fox, Grinshpun, Liebenau, Person and Szabo imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.
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 $G$ cont
ains 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.
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 additiona
lly 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.
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 $s_q(H_1,ldots,H_q)$ de
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