An Extension of Richardson’s Theorem in m-Colored Digraphs

Abstract

A digraph $D$ is said to be an $m$-coloured digraph if its arcs are coloured with $m$ colors. A directed path is called monochromatic if all of its arcs are coloured alike. A set $N \subseteq$ V ($D$) of vertices of $D$ is said to be a kernel by monochromatic paths of the $m$-coloured digraph $D$ if it satisfies the two following properties : (1) for any two different vertices $x$, $y$ $\in$ N there is no monochromatic directed path between them, and (2) for each vertex $u$ $\in$ V($D$)$\setminus N$ there exists a $uv$-monochromatic directed path, for some $v$ $\in N$. Let $D$ be an arc-coloured digraph. In 2009 Galeana-Sánchez introduced the concept of color-class digraph of $D$, denoted by $\fancyscript{C}_C$($D$), as follows: the vertices of the color-class digraph are the colors represented in the arcs of $D$, and ($i$, $j$) $\in$ $A$($\fancyscript{C}_C$($D$)) if and only if there exist two arcs namely ($u$, $v$) and ($v$, $w$) in $D$ such that ($u$, $v$) has color $i$ and ($v$, $w$) has color $j$. Galeana-Sánchez introduced the concept of color-class digraph mainly to prove that if $D$ is a fnite arc-coloured digraph such that $\fancyscript{C}_C$($D$) is a bipartite digraph, then D has a kernel by monochromatic paths. In this paper we will see some results related to the above result, which show the existence of kernels by monochromatic paths in arc-coloured digraphs. In particular, we will prove the following generalization of the previous result: Let $D$ be an $m$-coloured digraph and $\fancyscript{C}_C$($D$) its color-class digraph. If $\fancyscript{C}_C$($D$) has no cycles of odd length at least 3, then $D$ has a kernel by monochromatic paths. Finally we will prove Richardson’s Theorem as a direct consequence of the previous result.