CIS d-图的Δ-猜想
The $Δ$-Conjecture for CIS $d$-Graphs
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中文总结 AI 辅助
本文证明了源自Gurvich1978年博士论文的CIS d-图的Δ-猜想，明确了完全图边着色后各颜色子图极大稳定集交集非空时，着色不含彩虹三角形的结论。
中文摘要 AI 辅助
我们证明了可追溯至Gurvich1978年博士论文的Δ-猜想。具体而言，设完全图的边用颜色1,…,d着色，对每个i，令G_i为颜色i的边构成的图。我们证明：若对每个i∈[d]选取G_i的一个极大稳定集S_i，所有S_i的交集非空，则该着色不含彩虹三角形。
英文摘要
We prove the $Δ$-conjecture, which dates back to Gurvich's 1978 thesis. Specifically, let the edges of a complete graph be colored with colors $1,\ldots,d$, and for each $i$ let $G_i$ be the graph on the same vertex set formed by the edges of color $i$. We prove that if every choice of a maximal stable set $S_i$ of $G_i$, one for each $i\in[d]$, has nonempty intersection, then the coloring contains no rainbow triangle. Together with a result of Andrade, Boros, and Gurvich, this characterizes CIS $d$-graphs as precisely the Gallai $d$-graphs whose chromatic components are ordinary CIS graphs. We also show that every factor in the canonical modular decomposition of a CIS $d$-graph is a CIS $d$-graph whose edge-coloring uses at most two colors.
