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 $G$ be a nontrivial connected graph of order $n$ with an edge-coloring$c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges maybe colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if notwo adjacent edges of it are assigned the same color. Let $k$ be a fixedinteger with $2\leq k\leq n$. For a vertex subset $S\subseteq V(G)$ with$|S|\geq 2$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$ . A\emph{$k$-proper coloring} of $G$ is an edge-coloring of $G$ having theproperty that for every set $S$ of $k$ vertices of $G$, there exists a proper$S$-tree $T$ in $G$. The minimum number of colors that are needed in a$k$-proper coloring of $G$ is defined as the \emph{$k$-proper index} of $G$,denoted by $px_k(G)$. In this paper, we determine the 3-proper index of allcomplete bipartite and complete multipartite graphs and partially determine the$k$-proper index of them for $k\geq 4$.

The $k$-proper index of complete bipartite and complete multipartite graphs