It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan-Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another...

Alon and Tarsi presented in a previous paper a certain weighted sum over the set of all proper $k$-colorings of a graph, which can be computed from its graph polynomial. The subject of this paper is another combinatorial interpretation of the same quantity, expressed in terms of the numbers of certain modulo $k$ flows in the graph. Some relations between graph parameters can be obtained by combining these two formulas. For example: The number of proper 3-colorings of a 4-regular graph and the number...

If $G$ is a connected graph of order $n\ge 1$, then by a hamiltonian coloring of $G$ we mean a mapping $c$ of $V\left(G\right)$ into the set of all positive integers such that $|c\left(x\right)-c\left(y\right)|\ge n-1-D_G(x,y)$ (where $D_G(x,y)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x,y\in V\left(G\right)$. Let $G$ be a connected graph. By the hamiltonian chromatic number of $G$ we mean $$min(max(c\left(z\right);z\in V\left(G\right)\left)\right),$$ where the minimum is taken over all hamiltonian colorings $c$ of $G$. The main result of this paper can be formulated as follows: Let $G$ be a connected graph of order $n\ge 3$. Assume that there exists a subgraph...

An incidence in a graph G is a pair (v, e) with v ∈ V (G) and e ∈ E(G), such that v and e are incident. Two incidences (v, e) and (w, f) are adjacent if v = w, or e = f, or the edge vw equals e or f. The incidence chromatic number of G is the smallest k for which there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n = Cm2Cn equals 5 when...

For a nontrivial connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u-v$ path in $G$. Detour distance is a metric on the vertex set of $G$. For each integer $k$ with $1\le k\le n-1$, a coloring $c:V\left(G\right)\to \mathbb{N}$ is a $k$-metric coloring of $G$ if $\left|c\right(u)-c(v\left)\right|+D(u,v)\ge k+1$ for every two distinct vertices $u$ and $v$ of $G$. The value ${\chi }_m^k\left(c\right)$ of a $k$-metric coloring $c$ is the maximum color assigned by $c$ to a vertex of $G$ and the $k$-metric chromatic number ${\chi }_m^k\left(G\right)$ of $G$ is the minimum value of a $k$-metric coloring of $G$. For every...

Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V with f(v) = Ø the condition ∪u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. The weight of a kRDF f is the value w(f) = ∑v∈V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF...

A vertex coloring of a graph $G$ is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum $k$ for which $G$ has a multiset $k$-coloring is the multiset chromatic number ${\chi }_m\left(G\right)$ of $G$. For every graph $G$, ${\chi }_m\left(G\right)$ is bounded above by its chromatic number $\chi \left(G\right)$. The multiset chromatic number is determined for every complete multipartite graph as well as for cycles and their squares, cubes, and fourth powers. It is conjectured that for each $k\ge 3$, there exist sufficiently...

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G.We first define the operation μ(G) and then show that ω(μ(G)) = ω(G) and χ(μ(G)) = χ(G) + 1. This is done for arbitrary graph G, see also [10]....

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)-coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic number of an arbitrary graph.