arXiv subjects

Yanli Hao

Publications and source records attributed to Yanli Hao.

5 recordsLinked to original sources

arXiv2026

Improved bounds for the chromatic index of $k$-uniform hypergraphs

In 1997, Alon and Kim conjectured that if $H$ is a $k$-uniform $t$-simple hypergraph with maximum degree $D$ sufficiently large, then the chromatic index $\chi'(H)$ is upper bounded by $(t-1+1/t+\varepsilon)D$. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a $k$-uniform $t$-simple hypergraph $H$ with large maximum degree $D$ satisfies $$\chi'(H) \le (b+\varepsilon)kD,$$ where $b$ is a particular parameter derived from local structural information about $H$. We use structural techniques to prove sharp upper bounds on $b$ in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large $D$, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most $D$ has chromatic index at most $2.3581D$ and $2.6791D$, respectively.

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arXiv2026

Strong chromatic index of bipartite graphs

An edge-coloring of a graph $G$ is called a strong edge-coloring if all its color classes are induced matchings in $G$; the minimum number of colors required for such a coloring, denoted by $\chi_{s}'(G)$, is known as the strong chromatic index of $G$. For each vertex $v$ of a graph $G$, let $d_G(v)$ denote the degree of $v$ in $G$. Let $G$ be a bipartite graph with partite sets $A$ and $B$, and let $\Delta_A=\max\{d_G(a): a\in A\}$ and $\Delta_B=\max\{d_G(b): b\in B\}$. A conjecture of Brualdi and Quinn Massey asserts that $ \chi_s'(G) \le \Delta_A \Delta_B$. In this paper, we show that $\chi_s'(G) \le 1.676\, \Delta_A \Delta_B$ provided that the product $\Delta_A\Delta_B$ is sufficiently large.

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arXiv2024

A short proof of the Goldberg-Seymour conjecture

For a multigraph $G$, $χ'(G)$ denotes the chromatic index of $G$, $Δ(G)$ the maximum degree of $G$, and $Γ(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{ odd}\right\}$. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that $χ'(G)=\max\{Δ(G), Γ(G)\}$ or $χ'(G)=\max\{Δ(G) + 1, Γ(G)\}$. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long); and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an $O(|V|^5|E|^3)$ time algorithm for finding a $\max\{Δ(G) + 1, Γ(G)\}$-edge-coloring of $G$.

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arXiv2023

Decreasing the mean subtree order by adding $k$ edges

The mean subtree order of a given graph $G$, denoted $μ(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $μ(H) > μ(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $μ(H) < μ(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $μ(H) < μ(G)$. Furthermore, they proposed that $μ(K_m+nK_1) < μ(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures.

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arXiv2022

Linear arboricity of degenerate graphs

A linear forest is a union of vertex-disjoint paths, and the linear arboricity of a graph $G$, denoted by $\operatorname{la}(G)$, is the minimum number of linear forests needed to partition the edge set of $G$. Clearly, $\operatorname{la}(G) \ge \lceilΔ(G)/2\rceil$ for a graph $G$ with maximum degree $Δ(G)$. On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\operatorname{la}(G) \leq \lceil(Δ(G)+1) / 2\rceil$ for every graph $ G $. This conjecture has been verified for planar graphs and graphs whose maximum degree is at most $ 6 $, or is equal to $ 8 $ or $ 10 $. Given a positive integer $k$, a graph $G$ is $k$-degenerate if it can be reduced to a trivial graph by successive removal of vertices with degree at most $k$. We prove that for any $k$-degenerate graph $G$, $\operatorname{la}(G) = \lceilΔ(G)/2 \rceil$ provided $Δ(G) \ge 2k^2 -k$.

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