Tight bounds on the complexity of semi-equitable coloring of cubic and subcubic graphs

Abstract

A k-coloring of a graph G = (V, E) is called semi-equitable if there exists a partition of its vertex set into independent subsets V 1 , . . . , V k in such a way thatThe color class V 1 is called non-equitable. In this note we consider the complexity of semi-equitable k-coloring, k ≥ 4, of the vertices of a cubic or subcubic graph G. In particular, we show that, given a n-vertex subcubic graph G and constants > 0, k ≥ 4, it is NP-complete to obtain a semi-equitable k-coloring of G whose non-equitable color class is of size s if s ≥ n/3 + n, and it is polynomially solvable if s ≤ n/3. *

Population-based gradient descent weight learning for graph coloring problems

“…The GCP and the ECP have been studied very intensively in the past. Both problems are known to be NP-hard in the general case (see [20] for the GCP and [16] for the ECP). Thus, assuming that N = N P , no algorithm can solve these problems exactly in polynomial time except for specific cases.…”

Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

“…Chen and Yen (2012); Kierstead and Kostochka (2012); Kostochka and Nakprasit (2003); Kostochka et al (2005)), analysis of the problem's complexity (cf. Furmanczyk and Kubale (2018b)), designing exact algorithms for polynomial cases (cf. Kierstead et al (2010)), and approximation algorithms or heuristics for hard cases (cf.…”

The 3-Rainbow Domination Number of the Cartesian Product of Cycles

“…Kang et al [17] initiated the study of outer-independent k-rainbow domination and presented sharp lower and upper bounds on the outer-independent two-rainbow domination number. There are also some references related to k-coloring of a graph [18,19]. G 1 2G 2 , the Cartesian product of G 1 and G 2 , is the graph with the vertex set V(G 1 ) × V(G 2 ), and (u, v)(u , v ) ∈ E(G 1 2G 2 ) if either uu ∈ E(G 1 ) and v = v or vv ∈ E(G 2 ) and u = u .…”

“…Kang et al [17] initiate the study of outer-independent k-rainbow domination and present sharp lower and upper bounds on the outer-independent 2-rainbow domination number. There are also some references related to k-coloring of a graph [18,19].…”