超Dirac图的彩虹连接$2$-着色
Rainbow connecting $2$-colorings of super-Dirac graphs
- HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）
- University of Pannonia（佩奇大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究最小度至少为一半的图的彩虹2-边着色，证明对接近完全平衡二部图或两个同阶团的图存在所需着色，并推广至更广图类，同时完整解决渐近版本并给出开放问题。
中文摘要 AI 辅助
设$G$是一个最小度满足$\delta(G)\ge|V(G)|/2$的图。我们能否将$G$的边染成红色和蓝色，使得每一对不相邻的顶点都由一条恰好包含一条红边和一条蓝边的路径连接？我们对此问题给出肯定回答，针对一类“接近”完全平衡二部图或两个同阶团的并集的图类。令人惊讶的是，我们的方法可推广到最小度略高于$|V(G)|/2$的更大图类。此外，我们完整地回答了该问题的渐近版本，证明每个满足$\delta(G)\ge(|V(G)|-1)/2$的图$G$都存在一种$2$-边着色，使得几乎所有顶点对都由彩虹路径连接。我们还提出了若干相关开放问题。
英文摘要
Let $G$ be a graph with minimum degree $δ(G)\ge|V(G)|/2$. Can we color the edges of $G$ with red and blue so that every pair of non-adjacent vertices is connected by a path consisting of exactly one red edge and one blue edge? We provide an affirmative answer to this question for a class of graphs that are ``close'' to a complete balanced bipartite graph or the disjoint union of two cliques of the same order. Surprisingly, our methods extend to a much broader class of graphs with minimum degree slightly above $|V(G)|/2$. Furthermore, we answer an asymptotic version of this question in full, proving that every graph $G$ satisfying $δ(G)\ge(|V(G)|-1)/2$ has a $2$-edge-coloring such that almost all pairs of vertices are connected by a rainbow path. In addition, we propose a number of related open problems.
