Full Cycle Extendability of Triangularly Connected Almost Claw-Free Graphs

Abstract

This paper generalizes the concept of locally connected graphs. A graph $G$ is triangularly connected if for every pair of edges $e_1,e_2\in E(G)$, $G$ has a sequence of $3$-cycles $C_1,C_2,\dots ,C_l$ such that $e_1\in C_1,e_2\in C_l$ and $E(C_i)\cap E(C_{i+1})\ne \mathrm{\varnothing }$ for $1\le i\le l-1$. In this paper, we show that every triangularly connected $K_{1,4}$-free almost claw-free graph on at least three vertices is fully cycle extendable.