English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The research in the present paper was motivated by the conjecture ofRyj\'{a}\v{c}ek that every locally connected graph is weakly pancyclic.  For a connected locally connected graph $G$ of order at least $3$, ourresults are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ isweakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycleextendable 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 $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cupP_3)\}$-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$$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$ forevery induced path $vuw$ of order $3$ in $G$, then $G$ is fully cycleextendable, which implies that every connected locally Ore or locally Diracgraph of order at least $3$ is fully cycle extendable.

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