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

Abstract

An equitable coloring of a graph G = (V, E) is a (proper) vertex-coloring of G, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.

Approximation Algorithms for Graph Partition into Bounded Independent Sets

“…Many related studies have been conducted regarding conflict constraints, including parallel machine scheduling [7][8][9] to unrelated machine scheduling [10][11][12] . Several other studies focused on equitable coloring [13][14][15] , weighted coloring [16,17] , and subgraph partitioning [18] .…”

“…Suppose that jI 2 j < 4 15 n after all triangles compatible with e Y are put into I 2 . Since jY j < 5 15 n, jI 2 j < 4 15 n, and jY j > 13 15 n, the size of the children of e Y must exceed 4 15 n, contradicting jY n P k t D1 L 2t j < 4 15 n. This condition guarantees that at least 4 15 n triangles are included in I 2 . Since I 1 and I 2 contain at least 4 15 n triangles, we derive jI 3 j 7 15 n. Hence, Alg.T r / D maxfjI 1 j; jI 2 j; jI 3 jg 7 5 OP T .T r / holds.…”