Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

Ruonan Li,Shenggui Zhang,Hajo Broersma

doi:10.1007/s00373-018-1989-2

Ruonan Li, Shenggui Zhang + Show 1 more

Open Access

https://doi.org/10.1007/s00373-018-1989-2

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Abstract

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edge-colored complete graph to contain a small and a large properly edge-colored theta graph, respectively.

Highlights

All graphs considered in this paper are finite, simple, and undirected unless specified explicitly as directed graphs
In [14], it is revealed that if a colored complete graph G contains no monochromatic edge-cut and there exists a vertex v which is not contained in any PC cycles in G, a substructure of G is essentially a multipartite tournament
Our result indicates a possible approach to obtaining results on the existence of PC Hamilton cycles in colored complete graphs

Summary

Introduction

All graphs considered in this paper are finite, simple, and undirected unless specified explicitly as directed graphs. Definition 1.1 A colored complete graph G is essentially a multipartite tournament if there exists a mapping f : V (G) → col(G) such that col(uv) = f (u) or col(uv) = f (v) for each edge uv ∈ E(G). In [14], it is revealed that if a colored complete graph G contains no monochromatic edge-cut and there exists a vertex v which is not contained in any PC cycles in G, a substructure of G is essentially a multipartite tournament. Observation 1.1 clearly implies the following: if a colored complete graph G is essentially a multipartite tournament, G contains no PC theta graph. Our main result characterizes the difference between CD-critical colored complete graphs and essentially multipartite tournaments in terms of the (non)existence of PC theta graphs, in the following way.

Preliminaries

Colored Complete Graphs Without PC Theta Graphs

PC Theta Graphs of Small Order

PC Theta Graphs of Large Order

Concluding Remarks and Questions