Related papers: Perfect colourings of regular graphs
A {\it proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of…
A total colouring of a graph is a colouring of its vertices and edges such that no two adjacent vertices or edges have the same colour and moreover, no edge coloured $c$ has its endvertex coloured $c$ too. A weak total Thue colouring of a…
Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a…
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a…
We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be…
We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a…
A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…
We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its…
Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let $H$ be a (properly) edge-colored graph. The…
An $i$-independent set is a vertex set whose pairwise distance is at least $i+1$. A proper (square) $k$-coloring of a graph $G$ is a partition of its vertex set into $k$ independent ($2$-independent) sets. A packing $(1^{j}, 2^k)$-coloring…
If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote…
The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge…
Let $2\le k\in\mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an {\it efficient total coloring} if each color yields an efficient dominating set, where the efficient domination condition applies to the…
In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two…
Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…
An edge-colouring is {\em strong} if every colour class is an induced matching. In this work we give a formulae that determines either the optimal or the optimal plus one strong chromatic index of bipartite outerplanar graphs. Further, we…
A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup…
Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the…
Consider a vertex colouring game played on a simple graph with $k$ permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once…