arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1503.03283cs.DMmath.CO
收藏

完全二部图的无圈边染色

On Acyclic Edge-Coloring of Complete Bipartite Graphs

Ayineedi Venkateswarlu, Santanu Sarkar, A. Sai Mali

备注：
17 pages, 10 figures
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

图的无圈边染色是指不含双色圈的正常边染色。图$G$的无圈色指数，记为$a'(G)$，是使得$G$允许用$k$颜色进行无圈边染色的最小整数。设$Δ= Δ(G)$表示图$G$中顶点的最大度。每边都有$n$顶点的完全二部图记为$K_{n,n}$。Basavaraju，Kumran和Kummini证明了当$n$是奇数时，$a'(K_{n,n}) \ge n+2 = Δ+ 2$是奇数。Basavaraju和Rumran使用$p+2$颜色给出了$K_{p,p}$的非循环边着色，从而当$p$是奇素数时建立了$a'(K_{p,p}) = p+2 = Δ+ 2$。他们的方法中的主要工具是完美的$1$-$K_{p,p}$的因子分解。最近，Venkateswarlu和Sarkar遵循他们的方法证明了$K_{2p-1,2p-1}$允许使用$2p+1$颜色的非循环边着色，这意味着$a'(K_{2p-1,2p-1}) = 2p+1 = Δ+ 2$，其中$p$是奇素数。在本文中，我们推广了这种方法，并提出了一个一般的框架，可能得到一个非循环边着色的$K_{n,n}$，具有一个完美的$1$ -因子分解使用$n+2 = Δ+2$颜色。在这个一般的框架中，我们表明，$K_{p^2,p^2}$承认一个非循环边着色使用$p^2+2$颜色，从而建立$a'(K_{p^2,p^2}) = p^2+2 = Δ+ 2$时，$p\ge 5$是一个奇素数。

英文摘要：

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $Δ= Δ(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = Δ+ 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = Δ+ 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = Δ+ 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = Δ+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = Δ+ 2$ when $p\ge 5$ is an odd prime.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑