Given a digraph D=(V(D),A(D)), let ∂D+(v)={vw|w∈ND+(v)} and ∂D−(v)={uv|u∈ND−(v)} be semi-cuts of v. A mapping φ:A(D)→[k] is called weak-odd k-edge coloring D if it satisfies the condition: for each v∈V(D), there at least one color with an odd number occurrences on non-empty semi-cut We call minimum integer k chromatic index D. When limit to 2 colors, def(D) denote defect D, i.e vertices in whic...

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, are assigned different colors. The strong chromatic index of G is the smallest number k for which G has a strong k-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree tw...

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D′(G). We prove that D′(G) 6 D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chrom...