A note on compact and compact circular edge-colorings of graphs

Dariusz Dereniowski,Adam Nadolski

doi:10.46298/dmtcs.431

Abstract

Graphs and Algorithms We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

Highlights

In the paper we consider two variants of edge-coloring of weighted graphs
We prove that a compact circular coloring exists for every weighted odd cycle and provide an approximation algorithm
In the first part of this section we provide an algorithm for compact circular coloring of odd cycles

Summary

Introduction

In the paper we consider two variants of edge-coloring of weighted graphs. Let (G, w) be a weighted graph, where G = (V (G), E(G)) and w : E(G) → N. These notions motivate us to transform the definition of edge-coloring in such a way that arcs of a circle, instead of intervals, are assigned to the edges. If we multiplied by M the endpoints of all the colors assigned by c, we would obtain a standard or circular (M · r)-coloring This transformation would preserve the compactness property in both coloring models. In the papers [11, 12] the authors consider compact circular coloring of bipartite graphs in the context of application of this coloring model in scheduling, in the cyclic version of open shop. (Sec. 2) we discuss the relations between compact and circular compact colorings These considerations yield a characterization of the graphs which admit a standard compact coloring with arbitrary weights on the edges. We show that joining a path P2 to a vertex of a cycle makes the problem of colorability NP-hard

Relations between Standard and Circular Compact Colorings

Compact Circular Coloring of Odd Cycles

Concluding Remarks