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Register a new userLarge chromatic number and Ramsey graphs
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Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.
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A signed graph is a pair $(G, sigma)$, where $G$ is a graph and $sigma: E(G) to {+, -}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, sigma)$ a circular $r$-coloring of $(G, sigma)$ is an assignment $psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $sigma(e)=+$, then $psi(u)$ and $psi(v)$ have distance at least $1$, and if $sigma(e)=-$, then $psi(v)$ and the antipodal of $psi(u)$ have distance at least $1$. The circular chromatic number $chi_c(G, sigma)$ of a signed graph $(G, sigma)$ is the infimum of those $r$ for which $(G, sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $max{chi_c(G, sigma): sigma text{ is a signature of $G$}}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $chi_c(G, sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of M{a}v{c}ajov{a}, Raspaud, and v{S}koviera.
Resolving a problem raised by Norin, we show that for each $k in mathbb{N}$, there exists an $f(k) le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k) = O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we also show that for each $k in mathbb{N}$, there exists an $f_ell(k) le 4k$ such that every graph $G$ with list chromatic number at least $f_ell(k)+1$ contains a subgraph $H$ with both connectivity and list chromatic number at least $k$. This result is again best-possible up to multiplicative constants; here, unlike with $f(cdot)$, even the existence of $f_ell(cdot)$ appears to have been previously unknown.
By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]={1,2,ldots, n}$ for some integers $nge kge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in advance. Examples of such graphs have been investigated in a great variety of contexts in the literature with particular attention being paid to their chromatic number. In recent joint work with Tomasz {L}uczak, two of the authors embarked on a systematic study of the chromatic numbers of such type-graphs, formulated a general conjecture determining this number up to a multiplicative factor, and proved various results of this kind. In this article we fully prove this conjecture.
Let $G$ be a simple graph with maximum degree $Delta(G)$ and chromatic index $chi(G)$. A classic result of Vizing indicates that either $chi(G )=Delta(G)$ or $chi(G )=Delta(G)+1$. The graph $G$ is called $Delta$-critical if $G$ is connected, $chi(G )=Delta(G)+1$ and for any $ein E(G)$, $chi(G-e)=Delta(G)$. Let $G$ be an $n$-vertex $Delta$-critical graph. Vizing conjectured that $alpha(G)$, the independence number of $G$, is at most $frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $alpha(G)<frac{3n}{5}$. We show that for any given $varepsilonin (0,1)$, there exist positive constants $d_0(varepsilon)$ and $D_0(varepsilon)$ such that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $alpha(G)<(frac{{1}}{2}+varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d$ and $Delta(G)ge (d+2)^{5d+10}$, then [ alpha(G) < left. begin{cases} frac{7n}{12}, & text{if $d= 3$; } frac{4n}{7}, & text{if $d= 4$; } frac{d+2+sqrt[3]{(d-1)d}}{2d+4+sqrt[3]{(d-1)d}}n<frac{4n}{7}, & text{if $dge 19$. } end{cases} right. ]
The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that the size-Ramsey number of the grid graph on $ntimes n$ vertices is bounded from above by $n^{3+o(1)}$.
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