Integer flows on triangularly connected signed graphs

Abstract

A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1,T_2,\dots ,T_m$ in $G$ such that for any $i,j$ with $1\le i<j\le m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\varnothing$ if $j>i+1$. A connected graph $G$ is triangularly connected if for any two nonparallel edges $e$ and $e^{\prime }$ there is a triangle-path $T_1{T}_2\cdots T_m$ such that $e\in E(T_1)$ and $e^{\prime }\in E(T_m)$. For ordinary graphs, Fan {\it et al.}~(J. Combin. Theory Ser. B 98 (2008) 1325-1336) characterize all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result include integer flow of some families of ordinary graphs, such as, locally connected graphs due to Lai (J. Graph Theory 42 (2003) 211-219) and some types of products of graphs due to Imrich et al.(J. Graph Theory 64 (2010) 267-276). In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that every flow-admissible triangularly connected signed graph admits a nowhere-zero $4$-flow if and only if it is not the wheel $W_5$ associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere-zero $4$-flow but not $3$-flow.