模色指数的线性下界
Linear Lower Bounds for the Modular Chromatic Index
AI总结：
该研究反驳了关于图的模色指数的已有猜想，通过构造二分图给出了模色指数的线性下界，证明其下界可达$3k/2$量级。
AI中文摘要：
设$k\geq2$为整数，图$G$的$1\bmod k$边着色是指每个颜色类中每个非零度数都模$k$余1的边着色。设$\chi'_k(G)$为所需的最少颜色数，$\chi'_k$为所有有限简单图$G$的$\chi'_k(G)$的上确界。Botler、Colucci和Kohayakawa猜想存在绝对常数$C$，使得对所有$k$和所有$G$都有$\chi'_k(G)\leq k+C$。我们反驳了该猜想，甚至在二分图类中也成立。更准确地说，对所有整数$c\geq0$和$k\geq3c+2$，我们构造了有限简单二分图$G_{k,c}$，满足$\chi'_k(G_{k,c})=k+c+1$，因此对所有$k\geq2$有$\chi'_k\geq k+\lfloor(k+1)/3\rfloor$。对$k_m=2\cdot3^{m-1}$，我们给出有限简单二分图$G_m$的仿射超平面构造，满足$\Delta(G_m)=\chi'_{k_m}(G_m)=3^m=3k_m/2$。更一般地，对所有足够大的$k$，我们构造有限简单二分图$G_k$，满足$\Delta(G_k)=\chi'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$。我们的证明结合了共度数障碍、显式循环与仿射几何构造以及结构化随机扰动。
英文摘要：
Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $χ'_k(G)$ denote the minimum number of colors required, and let $χ'_k$ be the supremum of $χ'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $χ'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $χ'_k(G_{k,c})=k+c+1$. Consequently, $χ'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $Δ(G_m)=χ'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $Δ(G_k)=χ'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.
