Weak-odd chromatic index of special digraph classes

Abstract

Given a digraph D = ( V ( D ) , A ( D ) ) , let ∂ D + ( v ) = { v w | w ∈ N D + ( v ) } and ∂ D − ( v ) = { u v | u ∈ N D − ( v ) } be semi-cuts of v . A mapping φ : A ( D ) → [ k ] is called a weak-odd k -edge coloring of D if it satisfies the condition: for each v ∈ V ( D ) , there is at least one color with an odd number of occurrences on each non-empty semi-cut of v . We call the minimum integer k the weak-odd chromatic index of D . When limit to 2 colors, let def ( D ) denote the defect of D , i.e the minimum number of vertices in D at which the above condition is not satisfied. In this paper, we give a descriptive characterization with respect to the weak-odd chromatic index and the defect of semicomplete digraphs and extended tournaments, which generalize results of tournaments to broader classes. In addition, we initiate the study of weak-odd edge covering on digraphs.