Search

A matching $M$ in a graph $G$ is {\em semistrong} if every edge of $M$ has an endvertex of degree one in the subgraph induced by the vertices of $M$. A {\em semistrong edge-coloring} of a graph $G$ is a proper edge-coloring in which every color class induces a semistrong matching. In this paper, we continue investigation of properties of semistrong edge-colorings initiated by Gy\'{a}rf\'{a}s and Hubenko ({Semistrong edge coloring of graphs}. \newblock {\em J. Graph Theory}, 49 (2005), 39--47). We establish tight upper bounds for general graphs and for graphs with maximum degree $3$. We also present bounds about semistrong edge-coloring which follow from results regarding other, at first sight non-related, problems. We conclude the paper with several open problems.

A $d$-subsequence of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence, for $1 \le d \le k$, is non-repetitive, i.e. it contains no consecutive equal subsequences. In 2002, Grytczuk proposed a conjecture that for any $k$, $k+2$ symbols are enough to construct a $k$-Thue sequences of arbitrary lengths. So far, the conjecture has been confirmed for $k \in \{1,2,3,5\}$. Here, we present two different proving techniques, and confirm it for all $k$, with $2 \le k \le 8$.

{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvo\v{r}\'{a}k et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).

\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.

A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the maximum degrees of the two partite sets $2$ and $\Delta$ admits a strong edge coloring with at most $\Delta+2$ colors. We prove that this conjecture holds for such graphs with $\Delta=3$. We also confirm the conjecture proposed by Faudree et al. for subcubic bipartite graphs.