Equitable Coloring of Trees

Equitable coloring of random graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ=(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of ...

Vertex Equitable Labelings of Transformed Trees

Equitable List Coloring of Graphs

A graph G is equitably k-choosable if, for any k-uniform list assignment L, G admits a proper coloring π such that π(v) ∈ L(v) for all v ∈ V (G) and each color appears on at most |G|/k vertices. It was conjectured in [8] that every graph G with maximum degree ∆ is equitably k-choosable whenever k ≥ ∆ + 1. We prove the conjecture for the following cases: (i) ∆ ≤ 3; (ii) k ≥ (∆ − 1). Moreover, eq...

Equitable defective coloring of sparse planar graphs

A graph has an equitable, defective k-coloring (an ED-k-coloring) if there is a kcoloring of V (G) that is defective (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). A graph may have an ED-kcoloring, but no ED-(k + 1)-coloring. In this paper, we prove that planar graphs with minimum degree at least 2 and girth a...

On Equitable Coloring of d-Degenerate Graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most 1. A d-degenerate graph is a graph G in which every subgraph has a vertex with degree at most d. A star Sm with m rays is an example of a 1-degenerate graph with maximum degree m that needs at least 1 + m/2 colors for an equitable coloring. Our main result is that every n-...