Short Cycle Covers of Graphs with at Most 77% Vertices of Degree Two

Anna Kompišová ,Robert Lukoťka

doi:10.37236/9284

Abstract

Let $G$ be a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two and let $cc(G)$ be the length of its shortest cycle cover. It is known that if $cc(G) < 1.4m$ in bridgeless graphs with $n_2 \le m/10$, then the Cycle Double Cover Conjecture holds. Fan (2017) proved that if $n_2 = 0$, then $cc(G) < 1.6258m$ and $cc(G) < 1.6148m$ provided that $G$ is loopless; morever, if $n_2 \le m/30$, then $cc(G) < 1.6467m$. We show that for a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two, $cc(G) < 1.6148m + 0.0741n_2$. Therefore, if $n_2=0$, then $cc(G) < 1.6148m$ even if $G$ has loops; if $n_2 \le m/30$, then $cc(G) < 1.6173m$; and if $n_2 \le m/10$, then $cc(G) < 1.6223|E(G)|$. Our improvement is obtained by randomizing Fan's construction.

Highlights

We show that for a bridgeless multigraph with m edges and n2 vertices of degree two, cc(G) < 1.6148m + 0.0741n2
A cycle is a graph with all vertices of even degree and a circuit is an inclusion-wise minimal nonempty cycle
Let us contract all trees which we have introduced in the beginning the proof

Summary

Introduction

A cycle is a graph with all vertices of even degree and a circuit is an inclusion-wise minimal nonempty cycle. Let G be a bridgeless graph with m edges and n2 vertices of degree 2. Let G be a bridgeless graph (loops allowed) with m edges and n2 vertices of degree 2. Let G be a bridgeless graph of minimum degree 3 which contains m non-loop edges and s loops. There exists a probability distribution over proper spanning cycles F of G such that for every edge e we have P (e ∈ F ) = 2/3 if it is a non-loop and P (e ∈ F ) = 1/3 if it is a loop. If we contract an edge of the new 2-circuit in Ge into a single vertex v, we get Gs. Otherwise, we replace v by a tree in such a way that the new vertices have degree 3 and Ge is bridgeless (Figure 1b).

To bound the number of components of

Thus the length of the cover is at most