Related papers: Minimal colorings for properly colored subgraphs i…

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G))…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty…

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the…

Let $G=(V(G), E(G))$ be a graph with maximum degree $\Delta$. For a subset $M$ of $E(G)$, we denote by $G[V(M)]$ the subgraph of $G$ induced by the endvertices of edges in $M$. We call $M$ a semistrong matching if each edge of $M$ is…

A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to \{1,2,\ldots,k\}$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge…

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above…

A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance…

In an undirected graph, a conflict-free coloring (with respect to open neighborhoods) is an assignment of colors to the vertices of the graph $G$ such that every vertex in $G$ has a uniquely colored vertex in its open neighborhood. The…

A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four…

A {\em restraint} on a (finite undirected) graph $G = (V,E)$ is a function $r$ on $V$ such that $r(v)$ is a finite subset of ${\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\em permitted} by $r$ if $c(v) \not\in r(v)$ for all…

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log…

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural…

For a graph $G$, by $\chi_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, $\chi_2(G)$ is the chromatic number…

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…

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…

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…

The anti-Ramsey number, $ar(G, H)$ is the minimum integer $k$ such that in any edge colouring of $G$ with $k$ colours there is a rainbow subgraph isomorphic to $H$, i.e., a copy of $H$ with each of its edges assigned a different colour. The…

Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz local lemma to show the following two results about colourings c of the edges of the complete graph K_n. If for each vertex v of K_n the colouring c assigns each…

We prove that every graph with circumference at most $k$ is $O(\log k)$-colourable such that every monochromatic component has size at most $O(k)$. The $O(\log k)$ bound on the number of colours is best possible, even in the setting of…