arXiv subjects

Xuli Qi

Publications and source records attributed to Xuli Qi.

8 recordsLinked to original sources

arXiv2026

Edge-chromatic $4$-critical graphs and Overfull Conjecture for graphs with maximum degree $4$

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A graph $G$ is called edge-chromatic $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)< \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\Delta(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.

math.CO↗

arXiv2026

On distance spectral radius of power hypertrees with given number of pendant paths of fixed length

The distance spectral radius of a connected hypergraph is the largest eigenvalue of the distance matrix of the hypergraph. A pendant path of length l with l greater than or equal to 1 in a hypergraph G at vertex v sub l plus 1 is a path consisting of vertices and edges v1 e1 v2 up to vl el v(l+1). The vertex v(l+1) has degree at least 2, vertex v1 has degree 1, and each vertex vi has degree 2 for i from 2 to l. Every vertex belonging to edge ei except vi and v(i+1) has degree 1 for all i from 1 to l. We find the unique hypertree that maximizes or minimizes the distance spectral radius among all r-th power hypertrees with m edges and k pendant paths of length l, where r is no less than 3, k and l are no less than 1, and the product of k and l is smaller than m.

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arXiv2026

The average distance of spanning trees in terms of independence number

Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v \in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu(G)$, is defined to be the average of all distances between all pairs of vertices in $G$, i.e., $\mu (G) = \binom{n}{2}^{-1} \sum_{\{u,v\} \subset V(G)}d_G(u,v)$. The problem of finding a spanning tree of minimum average distance is known to be NP-hard, so establishing an upper bound for the minimum average distance among all spanning trees is of particular interest. Mukwembi (J. Graph Theory, 2014) showed that if $G$ is a connected graph of order $n$ with independence number $\alpha$, where $n > 2 \alpha - 1$, then $G$ has a spanning tree $T$ such that $\mu(T) \le \alpha + 2$. In this paper, we first improve the upper bound to $\mu(T) < \alpha + 1$ for $\alpha \ge 1$, and then we find the bound could be further improved when $\alpha$ becomes larger, so a better upper bound \[ \mu(T) < \left\{ \begin{array}{ll} \alpha+1 & \hbox{if } 1\le\alpha\le 6,\\ \alpha+\frac12+\frac{4(\alpha-1)}{\alpha^2} & \hbox{if } \alpha\ge 7, \end{array} \right. \] is established later. In the end, we give a remark to indicate our new upper bound is best possible in the sense of asymptotics (when $n$ and $\alpha$ are large enough).

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arXiv2026

On the Hilton-Zhao vertex-splitting conjecture

Let $G$ be a simple graph with order $n$, maximum degree $\Delta(G)$, and chromatic index $\chi'(G)$, respectively. A graph $G$ is edge-chromatic critical if $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. Assume that $G$ is an $n$-vertex connected regular Class $1$ graph, and let $G^*$ be obtained from $G$ by splitting one vertex into two vertices. Hilton and Zhao in 1997 proposed the vertex-splitting conjecture: if $\Delta(G)>\frac{n}{3}$, then $G^*$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $\Delta(G)\ge\frac{3n}{4}$. In this paper, we confirm the conjecture for $\Delta(G) \ge\frac{2n-2}{3}$.

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arXiv2025

A new improvement to the Overfull Conjecture

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\delta(G)$ and chromatic index $\chi'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $\chi'(G)=\D(G)+1$ and $\chi'(H)\textless \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor n/2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\D$-critical graph $G$ with $\D(G)\textgreater\frac{n}{3}$ is overfull. The Overfull Conjecture has many implications, such as that it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\D(G)\textgreater\frac{n}{3}$, and implies several longstanding conjectures in the area of graph edge coloring. Recently, Cao, Chen, Jing and Shan (SIAM J. Discrete Math. 2022) verified the Overfull Conjecture for $\D(G)-7\delta(G)/4\ge (3n-17)/4$. In this paper, we improve it for $\D(G)-5\delta(G)/3\ge (2n-7)/3$.

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arXiv2023

Bounds for eccentricity-based parameters of graphs

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity indices, denoted by $\sigma_0(G)$, $\sigma_1(G)$, and $\sigma_2(G)$, respectively. These are defined by $\sigma_0(G)=\frac{1}{|V(G)|}\sum_{u\in V(G)}e_G(u)$, $\sigma_1(G)=\sum_{u\in V(G)}e_G^2(u)$, and $\sigma_2(G)=\sum_{uv\in E(G)}e_G(u)e_G(v)$. We study lower and upper bounds on these parameters among $n$-vertex connected graphs with fixed diameter, chromatic number, clique number, or matching number. Most of the bounds are sharp, with the corresponding extremal graphs characterized.

math.CO↗

arXiv2022

Precoloring extension of Vizing's Theorem for multigraphs

Let $G$ be a graph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta(G)+\mu(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, \Delta(G)+\mu(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge coloring of $G$. Gir\~{a}o and Kang verified this conjecture for distance-$9$ matchings. In this paper, we improve the required distance from $9$ to $3$ for multigraphs $G$ with $\mu(G) \ge 2$.

math.CO↗

arXiv2013

The Kirchhoff indices and the matching numbers of unicyclic graphs

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. It found considerable applications in a variety of fields. In this paper, we determine the minimum Kirchhoff index among the unicyclic graphs with fixed number of vertices and matching number, and characterize the extremal graphs.

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