A Code for m-Bipartite Edge-Coloured Graphs

Abstract

An (n + 1)-coloured graph $\left(\Gamma,\gamma\right)$ is said to be $m-bipartite$ if m is the maximum integer so that every m-residue of $\left(\Gamma,\gamma\right)$ (i.e. every connected subgraph whose edges are coloured by only m colours) is bipartite; obviously, every (n + 1)-coloured graph, with n $\geq$ 2, results to be m-bipartite for some m, with 2 $\leq$ m $\leq$ n + 1. In this paper, a numerical $code$ of length (2n \textminus{} m + 1) $\times$ q is assigned to each m-bipartite (n + 1)-coloured graph of order 2q. Then, it is proved that$any\; two\; such\; graphs\; have\; the\; same\; code\; if\; and\; only\; if\; they\; are\; colour-isomorphic$, i.e. if a graph isomorphism exists, which transforms the graphs one into the other, up to permutation of the edge-colouring. More precisely, if H is a given group of permutations on the colour set, we face the problem of algorithmically recognizing H-isomorphic coloured graphs by means of a suitable defi{}nition of H-code.