Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number $a{c}^{\text{'}}\left(P_n\right)$ as $\left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for $n\ge 9$ and have found the exact value of $a{c}^{\text{'}}\left(P_n\right)$ for $n=5,6,7,8$. Here we determine the exact values of $a{c}^{\text{'}}\left(P_n\right)$ for $n\ge 8$. They are $2p^2-6p+8$ for $n=2p$ and $2p^2-4p+6$ for $n=2p+1$. The exact value of the radio antipodal number $ac\left(P_n\right)$ for the path $P_n$ of order $n$ has been determined by Khennoufa and Togni in 2005 as $2p^2-2p+3$ for $n=2p+1$ and $2p^2-4p+5$ for $n=2p$. Although the value of $ac\left(P_n\right)$ determined there is correct, we found a mistake in the proof of the lower bound when $n=2p$ (Theorem $6$). However,...

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple graph and $E_G\left(v\right)$ denote the set of edges incident with a vertex $v$. A neighbor sum distinguishing (NSD) total coloring $\phi$ of $G$ is a proper total coloring of $G$ such that ${\sum }_{z\in E_G\left(u\right)\cup \left\{u\right\}}\phi \left(z\right)\ne {\sum }_{z\in E_G\left(v\right)\cup \left\{v\right\}}\phi \left(z\right)$ for each edge $uv\in E\left(G\right)$. Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $\Delta$ admits an NSD total $(\Delta +3)$-coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $\Delta \ge 11$ but without $5$-cycles by applying the Combinatorial Nullstellensatz.

We create and discuss several modifications to traditional graph coloring. In particular, we classify various notions of coloring in a proper hierarchy. We concentrate on grid graphs whose colorings can be represented by natural number entries in arrays with various restrictions.

In this paper we present theoretical and algorithmic results for the computation of lower bounds on the chromatic number of a weighted graph. In particular, we study different ways of a possible improvement of the lower bound offered by a maximum weighted clique. Based on our findings we devise new algorithms and show their performance on random graphs.

2000 Mathematics Subject Classification: 05C55.For a given graph G let V(G) and E(G) denote the vertex and the edge set of G respevtively. The symbol G e → (a1, …, ar) means that in every r-coloring of E(G) there exists a monochromatic ai-clique of color i for some i ∈ {1,…,r}. The edge Folkman numbers are defined by the equality Fe(a1, …, ar; q) = min{|V(G)| : G e → (a1, …, ar; q) and cl(G) < q}. In this paper we prove a new upper bound on the edge Folkman number Fe(3,5;13), namely Fe(3,5;13)...

In this paper, we show that the maximal number of minimal colourings of a graph with $n$ vertices and the chromatic number $k$ is equal to $k^{n-k}$, and the single graph for which this bound is attained consists of a $k$-clique and $n-k$ isolated vertices.

A graph $G=(V,E)$ is called improperly $(d_1,\cdots ,d_k)$-colorable if the vertex set $V$ can be partitioned into subsets $V_1,\cdots ,V_k$ such that the graph $G\left[V_i\right]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1\le i\le k$. In this paper, we mainly study the improper coloring of $1$-planar graphs and show that $1$-planar graphs with girth at least $7$ are $(2,0,0,0)$-colorable.

In this paper we characterize $k$-chromatic graphs without isolated vertices and connected $k$-chromatic graphs having a minimal number of edges.

Chartrand and Kronk in 1969 showed that there are planar graphs whose vertices cannot be partitioned into two parts inducing acyclic subgraphs. In this note we show that the same is true even in the case when one of the partition classes is required to be triangle-free only.

We generalize the methods of Esperet and Zhu [6] providing an upper bound for the game colouring number of squares of graphs to obtain upper bounds for the game colouring number of m-th powers of graphs, m ≥ 3, which rely on the maximum degree and the game colouring number of the underlying graph. Furthermore, we improve these bounds in case the underlying graph is a forest.