Equitable Coloring of IC-Planar Graphs with Girth <i>g</i> ≥ 7

An equitable <i>k</i>-coloring of a graph <i>G</i> is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable <i>k</i>-coloring of <i>G</i>, then the graph <i>G</i> is said to be equitably <i>k</i>-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>7</mn></mrow></semantics></math></inline-formula> is equitably <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>-colorable, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the maximum degree of <i>G</i>.