Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Block graphs - some general results and their equitable colorings

Hanna Furmańczyk, Vahan Mkrtchyan

Abstract

In this paper, we consider some general properties of block graphs as well as the equitable coloring problem in this class of graphs. In the first part we establish the relation between two structural parameters for general block graphs. We also give complete characterization of block graphs with given value of parameter $α_{\min}$. In the next part of the paper we confirm the hypothesis for some subclass of GLS block graphs in which the problem of EQUITABLE COLORING is unlikely to be polynomial time solvable. We give also an equitable $(n+2)$-algorithm for all GLS block graphs. As a by product we prove that the equitable chromatic spectrum for the subclass of GLS block graphs is gap-free.

Block graphs - some general results and their equitable colorings

Abstract

In this paper, we consider some general properties of block graphs as well as the equitable coloring problem in this class of graphs. In the first part we establish the relation between two structural parameters for general block graphs. We also give complete characterization of block graphs with given value of parameter . In the next part of the paper we confirm the hypothesis for some subclass of GLS block graphs in which the problem of EQUITABLE COLORING is unlikely to be polynomial time solvable. We give also an equitable -algorithm for all GLS block graphs. As a by product we prove that the equitable chromatic spectrum for the subclass of GLS block graphs is gap-free.
Paper Structure (7 sections, 24 equations, 1 figure)

This paper contains 7 sections, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A relation between $dc(G)$ and $\alpha_{\min}(G)$ in block graphs
  3. A characterization theorem for block graphs
  4. Equitable coloring of some block graphs
  5. Algorithm for equitable $t$-coloring of some GLS block graphs with $t\geq k+2$
  6. Equitable $(n+2)$-colorability of general GLS block graphs
  7. Conclusion and Future Work

Figures (1)

  • Figure 5: GLS block graph constructed for BIN-PACKING instance with $A=\{3,3,3,3\}$, $k=3$ and $B=4$gomes:par.

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof