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 $G=(V(G), E(G))$ be a graph with maximum degree $\Delta$. For a subset$M$ of $E(G)$, we denote by $G[V(M)]$ the subgraph of $G$ induced by theendvertices of edges in $M$. We call $M$ a semistrong matching if each edge of$M$ is incident with a vertex that is of degree 1 in $G[V(M)]$. Given apositive integer $k$, a semistrong $k$-edge-coloring of $G$ is an edge coloringusing at most $k$ colors in which each color class is a semistrong matching of$G$. The semistrong chromatic index of $G$, denoted by $\chi'_{ss}(G)$, is theminimum integer $k$ such that $G$ has a semistrong $k$-edge-coloring. Recently,Lu\v{z}ar, Mockov\v{c}iakov\'a and Sot\'ak conjectured that $\chi'_{ss}(G)\le\Delta^{2}-1$ for any connected graph $G$ except the complete bipartite graph$K_{\Delta,\Delta}$. In this paper, we settle this conjecture by proving thateach such graph $G$ other than a cycle on $7$ vertices has a semistrong edgecoloring using at most $\Delta^{2}-1$ colors.

Proving a conjecture on the upper bound of semistrong chromatic indices of graphs