Related papers: Separating Tree-chromatic number from Path-chromat…
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…
Resolving a 1985 open problem of Gy\'arf\'as, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving…
For a given graph $H$, the anti-Ramsey number of $H$ is the maximum number of colors in an edge-coloring of a complete graph which does not contain a rainbow copy of $H$. In this paper, we extend the decomposition family of graphs to the…
The Gy\'arf\'as-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an…
Duffy et al. [C. Duffy, G. MacGillivray, and \'E. Sopena, Oriented colourings of graphs with maximum degree three and four, Discrete Mathematics, 342(4), p. 959--974, 2019] recently considered the oriented chromatic number of connected…
The clustering of a graph coloring is the maximum size of monochromatic components. This paper studies colorings with bounded clustering in graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler…
The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a…
Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a…
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erd\" os, Simonovits and S\' os. For given graphs $G$ and $H$ the \emph{anti-Ramsey number} $\textrm{ar}(G,H)$ is defined to be the maximum number $k$…
The chromatic number $\chi\left(\mathcal{E^2}\right)$ of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on…
The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the…
For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…
We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast…
A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded…
We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric…
The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Gr\"unbaum (1973) together with the acyclic coloring of…
The $c$-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$-intersecting hypergraph has bounded $(t +…
Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed…