可分解多拟阵及其与图着色的联系

次立方图的局部不规则边染色

Locally irregular edge-coloring of subcubic graphs

本文研究次立方图的局部不规则边染色问题，证明最大度为3的可分解无爪图、循环置换图和广义Petersen图至多需要3种颜色，并给出需要3种颜色的围长4三次图无限族。

Partial coloring, vertex decomposability, and sequentially Cohen-Macaulay simplicial complexes

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

多拟阵之于有限群，犹如拟阵之于有限域

Polymatroids are to finite groups as matroids are to finite fields

本文定义有限群直积子群上的多拟阵 P(H,b)，推广有限域上拟阵理论，建立排除子式可表示性、Crapo-Rota 临界定理、对偶表示、Greene 定理与 MacWilliams 恒等式等对应关系，并将正常着色与无处零流的对偶性推广到超图。

Point partition numbers: decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number $χ_t(G)$ is the least integer $k$ for which $G$ admits a coloring with $k$ colors such that each color class induces a $(t-1)$-degenerate subgraph of $G$. So $χ_1$ is the chromatic number and $χ_2$ is the point aboricity. The point partition number $χ_t$ with $t\geq 1$ was introduced by Lick and White. A graph $G$ is called $χ_t$-critical if every proper subgraph $H$ of $G$ satisfies $χ_t(H)<χ_t(G)$. In this paper we prove that if $G$ is a $χ_t$-critical graph whose orde…

Families of trees decompose the random graph in any arbitrary way

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $α_1 e(H_1) + ... + α_k e(H_k) = m$ where each $α_i$ is a nonnegative integer, there is a coloring of the edges of $G$ with $α_1+...+α_k$ colors such that exactly $α_i$ color classes induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree, …

Odd edge-colorings of subdivisions of odd graphs

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum value of $k$ for which such a decomposition of $G$ exists is the odd chromatic index, $χ_{o}'(G)$, introduced by Pyber (1991). For every $k\geqχ_{o}'(G)$, the graph $G$ is said to be odd $k$-edge-colorable. Apart from two particular exceptions, which are respectively odd $5$- and odd $6$-edge-colorable, the rest of connected loopless graphs are odd $4$-edge-colo…