TY - JOUR
T1 - $\mathbb{Z}_3$-Connectivity of 4-Edge-Connected Triangular Graphs
AU - Zhou , Chuixiang
JO - Annals of Applied Mathematics
VL - 4
SP - 428
EP - 438
PY - 2022
DA - 2022/06
SN - 33
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/aam/20622.html
KW - $\mathbb{Z}_3$-connected, nowhere-zero 3-flow, triangular graphs.
AB - <p style="text-align: justify;">A graph $G$ is $k$-triangular if each of its edge is contained in at least $k$ triangles. It is conjectured that every 4-edge-connected triangular graph admits
a nowhere-zero 3-flow. A triangle-path in a graph G is a sequence of distinct
triangles $T_1T_2 · · · T_k$ in $G$ such that for $1 ≤ i ≤ k − 1,$ $|E(T_i) ∩ E(T_{i+1})| = 1$ and $E(T_i) ∩ E(T_j ) = ∅$ if $j &gt; i + 1.$ Two edges $e,$ $e′ ∈ E(G)$ are triangularly
connected if there is a triangle-path $T_1, T_2, · · · , T_k$ in $G$ such that $e ∈ E(T_1)$ and $e
′ ∈ E(T_k).$ Two edges $e,$ $e′ ∈ E(G)$ are equivalent if they are the same,
parallel or triangularly connected. It is easy to see that this is an equivalent
relation. Each equivalent class is called a triangularly connected component.
In this paper, we prove that every 4-edge-connected triangular graph $G$ is $\mathbb{Z}_3$-connected, unless it has a triangularly connected component which is not $\mathbb{Z}_3$-connected but admits a nowhere-zero 3-flow.</p>