Restrained star edge coloring of graphs and its application in optimal & safe storage practices

In this paper we introduce the concept of restrained star edge coloring of graphs by restraining the conditions of the star coloring of graphs. The restrained star edge coloring of graphs is a path based graph coloring which is said to be proper if all the bichromatic subgraphs of the graph are in the form of a galaxy. The minimum requirement for this coloring is its restrained star chromatic index, denoted as χ'rs. This paper exclusively explains, the restrained star edge coloring of several families of graphs including path, cycle, wheel, etc., and provides the exact value of its respective restrained star chromatic index, χ'rs with the usage of appropriate illustrations. In addition to this, an application of this coloring in the optimal utilization of storage spaces and in ensuring safe storage practices is also briefly elaborated

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page