Degree sum condition for $Z_3$-connectivity in graphs

Abstract

Let $G$ be a 2-edge-connected simple graph on $n$ vertices, let $A$ denote an abelian group with the identity element 0, and let $D$ be an orientation of $G$. The boundary of a function $f:E\left(G\right)\to A$ is the function $\partial f:V\left(G\right)\to A$ given by $\partial f\left(v\right)={\sum }_{e\in E^{+}\left(v\right)}f\left(e\right)-{\sum }_{e\in E^{-}\left(v\right)}f\left(e\right)$, where $E^{+}\left(v\right)$ is the set of edges with tail $v$ and $E^{-}\left(v\right)$ is the set of edges with head $v$. A graph $G$ is $A$-connected if for every $b:V\left(G\right)\to A$ with ${\sum }_{v\in V\left(G\right)}b\left(v\right)=0$, there is a function $f:E\left(G\right)\to A-\left\{0\right\}$ such that $\partial f=b$. In this paper, we prove that if $d\left(x\right)+d\left(y\right)\ge n$ for each $xy\in E\left(G\right)$, then $G$ is not $Z_3$-connected if and only if $G$ is either one of $15$ specific graphs or one of $K_{2,n-2},K_{3,n-3},K_{2,n-2}^{+}$ or $K_{3,n-3}^{+}$ for $n\ge 6$, where $K_{r,s}^{+}$ denotes the graph obtained from $K_{r,s}$ by adding an edge joining two vertices of maximum degree. This result generalizes the result in [G. Fan, C. Zhou, Degree sum and Nowhere-zero 3-flows, Discrete Math. 308 (2008) 6233–6240] by Fan and Zhou.