Almost eulerian compatible spanning circuits in edge-colored graphs

Zhiwei Guo,Hajo Broersma,Binlong Li,Shenggui Zhang

doi:10.1016/j.disc.2020.112174

Zhiwei Guo, Hajo Broersma + Show 2 more

Open Access

https://doi.org/10.1016/j.disc.2020.112174

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Abstract

Let G be a (not necessarily properly) edge-colored graph. A compatible spanning circuit in G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. As two extremal cases, the existence of compatible (i.e., properly edge-colored) Hamilton cycles and compatible Euler tours have been studied extensively. More recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex v of G at least ⌊(d(v)−1)∕2⌋ times in graphs satisfying Ore-type degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times. We respectively consider graphs satisfying Fan-type degree conditions, graphs with a high edge-connectivity, and the asymptotical existence of such compatible spanning circuits in random graphs.

Highlights

In this paper we consider only finite undirected graphs without loops or multiple edges
Using the proof technique applied in our proof of Corollary 1.1, we can confirm the existence of compatible spanning circuits visiting each vertex at least a specified number of times in edge-colored graphs with a high edgeconnectivity, as well as the asymptotical existence of compatible spanning circuits visiting each vertex v at least ⌊(d(v) − 1)/2⌋ times in edge-colored random graphs, as shown in the following results
We mainly considered the existence of compatible spanning circuits visiting each vertex at least a specified number of times in edge-colored graphs that satisfy Fan-type degree conditions

Summary

Introduction

In this paper we consider only finite undirected graphs without loops or multiple edges. By restricting the number of colors, Das and Rao [9] in 1983 considered the existence of more general compatible spanning circuits (i.e., not necessarily a compatible Hamilton cycle or Euler tour) in specific edge-colored graphs. In [9], they established necessary and sufficient conditions for the existence of compatible spanning circuits visiting each vertex exactly a specified number of times in 2-edge-colored complete graphs. Using the proof technique applied in our proof of Corollary 1.1, we can confirm the existence of compatible spanning circuits visiting each vertex at least a specified number of times in edge-colored graphs with a high edgeconnectivity, as well as the asymptotical existence of compatible spanning circuits visiting each vertex v at least ⌊(d(v) − 1)/2⌋ times in edge-colored random graphs, as shown in the following results. We will list the key ingredients for our proofs of the above results that are postponed to Section 3

Preliminaries

Proofs of the main results

Conclusions and final remarks