arXiv subjects
Publications and source records attributed to Watcharintorn Ruksasakchai.
In 1980, Athreya, Pranesachar and Singhi established the chromatic polynomial of $(3 \times n)$-Latin rectangles whose entries based on a set $\{1, 2, ..., λ\}$ in which $λ\geq n$. Their proof requires Möbius inversion formula and lattice partitions. In this paper, we present a simpler proof by using the idea of mathematical induction and appropriate coloring.
An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $q$-coloring of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r$ and the sizes of any two color classes differ by at most $1.$ Let the \emph{strong equitable vertex $r$-arboricity} of a graph $G,$ denoted by $va^\equiv_r (G)$, be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ The values of $va^\equiv_1 (K_{n,n})$ were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of $va^\equiv_1 (K_{n,n})$ were found in some special cases. In this paper, we extend their results by giving the exact values of $va^\equiv_1 (K_{n,n})$ for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each $va^\equiv_1 (K_{m,n})$ and $va^\equiv_1 (G)$ where $G$ is a balanced complete $k$-partite graph $K_{n,\ldots,n}.$
A {\em strong edge coloring} of a graph is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper, we prove that if $G$ is a graph with maximum degree at most four and maximum average degree less than $3$, then the list strong chromatic index is at most $3Δ+ 1$, where $Δ$ is the maximum degree of $G$. In addition, we prove that if $G$ is a planar graph with maximum degree at least $4$ and girth at least $7$, then the list strong chromatic index is at most $3Δ$.