Richardson’s Theorem for [formula omitted]-colored kernels in strongly connected digraphs

Abstract

A digraph D=(V,A) is said to be m-colored if its arcs are colored with m colors. An m-colored digraph D has a k-colored kernel if there exists K⊆V such that (i) for every v∈V∖K there exist a q-colored directed path, with q≤k, from v to a vertex of K, and (ii) for every pair {u,v}⊆K every directed path from u to v uses at least k+1 colors.Given an m-colored digraph D, the color-class digraph of D, denoted C(D), is defined as follows: the vertices of C(D) are the m colors of D, and (i,j) is an arc of C(D) if and only if there exist two consecutive arcs in D, namely (u,v) and (v,w), such that (u,v) has color i and (v,w) has color j.A digraph D is said to be cyclicallyk-partite if there is a partition {Vi}i=0k−1 of its vertices in independent sets such that every arc in D is either a loop or a ViVi+1-arc (taken the index modulo k). In Galeana-Sánchez (2012) it was proved that given an m-colored digraph D, if C(D) is cyclically 2-partite then D has a kernel by monochromatic paths (that is a 1-colored kernel). In this paper we extend this work and prove the following: LetDbe a strongly connectedm-colored digraphDsuch that, for somek≥1,C(D)is a cyclically(k+1)-partite digraph, with partition{Ci}i=0k. (i) If for some partCj, no vertex ofCjhas a loop, thenDhas ak-colored kernel. (ii) For eachi, with0≤i≤k, letDibe the subgraph ofDinduced by the set of arcs with color inCi, and for each vertexxofDletNC+(x)andNC−(x)be the set of colors appearing in the ex-arcs and in-arcs ofx, respectively. If for some subdigraphDj, for every vertexxofDjwe have thatNC+(x)⊈NC−(x), thenDhas ak-colored kernel.As a direct consequence we obtain a proof of Richardson’s Theorem in the case D is strongly connected, and a proof of a classical result by M. Kwaśnik (see Kwaśnik (1983) ) on the existence of k-kernels (a k-kernel of a digraph D=(V,A) is a set S⊆V such that for any v∈V∖S, dD(v,S)≤k−1 and for every pair {u,v}⊆S, dD(u,v)≥k) that asserts that if D is a strongly connected digraph such that every directed cycle has length congruent with 0 modulo k, then D has a k-kernel.