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/

Zendy Plus

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

Zendy Tools

Zendy Open

In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237]proved that if $G$ is a connected graph satisfying $\sigma_2(G) \geq |G|-k+1$then $G$ has a spanning $k-$ended tree. They also gave an example to show thatthe condition "$\sigma_2(G) \geq |G|-k+1$" is sharp. In this paper, weintroduce a new progress for this result. Let $K_{m,m+k}$ be a completebipartite graph with bipartition $V(K_{m,m+k})=A\cup B, |A|=m, |B|=m+k.$ Denoteby $H$ to be the graph obtained from $K_{m,m+k}$ by adding (or no adding) someedges with two end vertices in $A.$ We prove that if $G$ is a connected graphsatisfying $\sigma_2(G) \geq |G|-k$ then $G$ has a spanning $k-$ended treeexcept for the case $G$ is isomorphic to a graph $H.$ As a corollary of ourmain result, a sufficient condition for a graph to have a few branch verticesis given.

Progress on sufficient conditions for a graph to have a spanning $k-$ended tree