Related papers: Ore's Conjecture on color-critical graphs is almos…

The proper conflict-free chromatic number, $\chi_{pcf}(G)$, of a graph $G$ is the least $k$ such that $G$ has a proper $k$-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

For $k\in \mathbb{N}$, a $k$-acyclic colouring of a graph $G$ is a function $f\colon V(G)\to \{0,1,\dots,k-1\}$ such that (i)~$f(u)\neq f(v)$ for every edge $uv$ of $G$, and (ii)~there is no cycle in $G$ bicoloured by $f$. For $k\in…

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour…

Given an integer $r\gs 2$, an important theorem first proved by B. Andr\'asfai, P. Erd\H{o}s, and V. T. S\'os states that any $K_{r+1}$--free graph on $n$ vertices whose minimum degree is greater than $(3r-4)n/(3r-1)$ is $r$--colourable,…

Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$ to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in $K_n$, there…

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph…

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

An oriented graph $\overrightarrow{G}$ is pushably $k$-critical if it is not pushably $k$-colorable, but every proper subgraph of $\overrightarrow{G}$ is. The main result of this article is that every pushably $3$-critical oriented graph on…

Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A…

For $p\in \mathbb{N}$, a coloring $\lambda$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $\lambda$ or there is a color that appears exactly…

A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose…

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors…

A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential…

The chromatic edge-stability number ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $\chi(G')=\chi(G)-1$. Edge-stability critical graphs are introduced as the graphs…

Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for…