Related papers: Tomescu's graph coloring conjecture for $\ell$-con…

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the…

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$…

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 $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

An edge-colored graph $G$ is rainbow connected if every pair of vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of $G$ is defined to be the minimum integer $t$ such that there…

The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$…

For integers $k\ge1$ and $m\ge2$, let $g(k,m)$ be the least integer $n\ge1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. Refining the recent result…

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

For a finite graph $G$, we study the maximum $2$-edge colorable subgraph problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the matching number of $G$, and $\mu(G)$ is the size of the largest matching in any pair…

A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow…

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color can appear with less than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…

An edge-coloring of a connected graph $G$ is called a {\it monochromatic connection coloring} (MC-coloring, for short), introduced by Caro and Yuster, if there is a monochromatic path joining any two vertices of the graph $G$. Let $mc(G)$…

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…

A path in an edge-colored graph is called a \emph{rainbow path} if all edges on it have pairwise distinct colors. For $k\geq 1$, the \emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the minimum number of colors required…

For a graph $G$, let $\cn(G)$ and $\la(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \cite{Dirac53} implies that every graph $G$ satisfies $\cn(G)\leq \la(G)+1$. In…

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable…

A complete $k$-coloring of a graph $G$ is a (not necessarily proper) $k$-coloring of the vertices of $G$, such that each pair of different colors appears in an edge. A complete $k$-coloring is also called connected, if each color class…

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\mathcal{G}$ is said to be {\it$\chi$-bounded} if there exists…

In 1987, A. Gy\'arf\'as in his paper ``Problems from the world surrounding perfect graphs'' posed the problem of determining the smallest $\chi$-binding function for $\mathcal{G}(F,\overline{F})$, when $\mathcal{G}(F)$ is $\chi$-bounded. So…