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

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if forevery cutset $S$ of $G$, the ratio of $|S|$ to the number of components of$G-S$ is at least $t$. The Toughness Conjecture of Chv\'atal, stating thatthere exists a constant $t_0$ such that every $t_0$-tough graph with at leastthree vertices is hamiltonian, is still open in general. For any given integer$k\ge 1$, a graph $G$ is $(P_2\cup kP_1)$ free if $G$ does not contain thedisjoint union of $P_2$ and $k$ isolated vertices as an induced subgraph.  In this note, we show that every 4-tough and $2k$-connected $(P_2\cupkP_1)$-free graph with at least three vertices is hamiltonian.  This result in some sense is an "extension" of the classicalChv\'{a}tal-Erd\H{o}s Theorem that every $\max\{2,k\}$-connected  $(k+1)P_1$-free graph on at least three vertices is hamiltonian.

A note on hamiltonian cycles in $4$-tough $(P_2\cup kP_1)$-free graphs