Some Results on incidence coloring, star arboricity and domination number

TL;DR

This paper establishes inequalities linking incidence chromatic number, star arboricity, and domination number, providing bounds and specific results for various graph classes, including planar and cubic graphs.

Contribution

It introduces new inequalities connecting three graph invariants and derives bounds for the incidence chromatic number across different graph types.

Findings

Derived bounds for incidence chromatic number of all graphs.

Reduced upper bounds for planar graphs.

Cubic graphs with certain orders are not 4-incidence colorable.

Abstract

Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.