关于约翰逊图$J(n,2)$的消色指数
On the achromatic index of Johnson graphs $J(n,2)$
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中文总结 AI 辅助
本研究针对约翰逊图$J(n,2)$的消色指数问题，通过构造显式的正常完备边着色方案，给出该参数的新上下界，并确定多类参数取值下的精确结果。
中文摘要 AI 辅助
本文研究约翰逊图$J(n,2)$（也称为$n$-三角图）的正常完备边着色。这类图既同构于完全图的2-令牌图，也同构于完全图的线图。图$G$的$t$-边着色是指从$\{1,2,\ldots,t\}$中为每条边分配一种颜色的函数。若任意两条相邻边颜色均不相同，则称该着色为正常着色；若每对不同颜色都出现在某对相邻边上，则称该着色为完备着色。消色指数记为$α_2(G)$，是使得$G$存在正常完备$t$-边着色的最大整数$t$。我们建立了$α_2(J(n,2))$的新上下界，给出了达到下界的显式正常完备边着色方案，并确定了多类$n$取值下$α_2(J(n,2))$的精确值。
英文摘要
In this paper, we study proper and complete edge-colorings of Johnson graphs $J(n,2)$, also called $n$-triangular graphs. They are isomorphic both to the 2-token graphs of complete graphs and to the line graphs of complete graphs. A $t$-edge-coloring of a graph $G$ is a function that assigns one color from $\{1,2,\ldots,t\}$ to each edge. Such a coloring is called proper if no two incident edges receive the same color, and complete if every pair of distinct colors appears on a pair of incident edges. The achromatic index, denoted by $α_2(G)$, is the largest integer $t$ for which $G$ admits a proper and complete $t$-edge-coloring. We establish new lower and upper bounds for $α_2(J(n,2))$, provide explicit proper and complete edge-colorings attaining the lower bounds, and determine the exact value of $α_2(J(n,2))$ for several values of $n$.
