Related papers: Towards Cereceda's conjecture for planar graphs

We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours…

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k \geq 2$ and $r \geq k + \Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr} \omega (G) = \chi (G) = k, &…

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We…

We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and…

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

Let $c_1, c_2, \cdots, c_k$ be $k$ non-negative integers. A graph $G$ is $(c_1, c_2, \cdots, c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, \ldots, V_k$, such that the subgraph $G[V_i]$, induced by $V_i$, has…

We prove that for all integers $\Delta,r \geq 2$, there is a constant $C = C(\Delta,r) >0$ such that the following is true for every sequence $\mathcal{F} = \{F_1, F_2, \ldots\}$ of graphs with $v(F_n) = n$ and $\Delta(F_n) \leq \Delta$,…

For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$…

The page number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph…

Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We…

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a…

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say…

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in $O^*(2^n)$ time, as shown by Bj\"orklund, Husfeldt and Koivisto in 2009. For $k=3,4$, better algorithms are known for the $k$-coloring problem.…

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so…

We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the $k$-colouring graph $\mathcal{C}_{k}(G)$ of a graph $G$ correspond to the proper $k$-colourings of a graph $G$, with two…

We prove a conjecture of Dvo\v{r}\'ak, Kr\'al, Nejedl\'y, and \v{S}krekovski that planar graphs of girth at least five are square $(\Delta+2)$-colorable for large enough $\Delta$. In fact, we prove the stronger statement that such graphs…

A recolouring sequence, between $k$-colourings $\alpha$ and $\beta$ of a graph $G$, transforms $\alpha$ into $\beta$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The…

In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two…