Local Connectivity, Local Degree Conditions, some Forbidden Induced Subgraphs, and Cycle Extendability

TL;DR

This paper investigates conditions under which locally connected graphs are weakly pancyclic or fully cycle extendable, focusing on forbidden subgraphs and local degree conditions, thus advancing understanding of cycle properties in such graphs.

Contribution

It establishes new cycle extendability results for locally connected graphs with specific forbidden subgraphs and degree conditions, confirming conjectures and extending prior work.

Findings

Locally connected graphs without certain subgraphs are weakly pancyclic.

Certain forbidden subgraphs guarantee full cycle extendability.

Degree and neighborhood intersection conditions imply cycle extendability.

Abstract

The research in the present paper was motivated by the conjecture of Ryj\'{a}\v{c}ek that every locally connected graph is weakly pancyclic. For a connected locally connected graph $G$ of order at least $3$, our results are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is weakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycle extendable if and only if $2\delta(G)\geq n(G)$. If $G$ is $\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$-free or $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$-free, then $G$ is fully cycle extendable. If $G$ is distinct from $K_1+K_1+\bar{K}_3$ and $\{ K_1+P_4,K_{1,4},K_2+(K_1\cup K_2)\}$-free, then $G$ is fully cycle extendable. Furthermore, if $G$ is a connected graph of order at least $3$ such that $$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$ for every induced path $vuw$ of order $3$ in $G$, then $G$ is fully cycle extendable, which implies that every connected locally Ore or locally Dirac graph of order at least $3$ is fully cycle extendable.