An incidence of an undirected graph G is a pair (v, e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, (ii) e = f or (iii) vw = e or f . An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2012, Yang [15] proved that ev...

In this paper, the structural properties of 1-trees are discussed in details firstly. Based on the properties of 1-trees, the incidence chromatic number of 2-connected 1-trees can be determined.

Two inequalities are established connecting the graph invariants of incidence chromatic number, star arboricity and domination number. Using these, upper and lower bounds are deduced for the incidence chromatic number of a graph and further reductions are made to the upper bound for a planar graph. It is shown that cubic graphs with orders not divisible by four are not 4-incidence colorable. Sh...

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f , or (iii) vw = e or f . An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1, . . . , p} such that every two adjacent incidences receive distinct colours. In...

We introduce the incidence game chromatic number which unifies the ideas of game chromatic number and incidence coloring number of an undirected graph. For kdegenerate graphs with maximum degree ∆, the upper bound 2∆ + 4k − 2 for the incidence game chromatic number is given. If ∆ ≥ 5k, we improve this bound to the value 2∆ + 3k − 1. We also determine the exact incidence game chromatic number of...