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Mingxian Zhong

Publications and source records attributed to Mingxian Zhong.

13 recordsLinked to original sources
arXiv2025

3-Coloring $P_t$-Free Graphs With Only One Prescribed Induced Odd Cycle Length

A graph is $P_t$-free if it contains no induced subgraph isomorphic to a $t$-vertex path. A graph is not bipartite if and only if it contains an induced subgraph isomorphic to a $k$-vertex cycle, where $k$ is odd. We focus on the 3-coloring problem for $P_t$-free graphs that have only one prescribed induced odd cycle length. For any integer $t$ and any odd integer $k$, let $\mathcal{G}_{t,k}$ be the class of graphs that are $P_{t}$-free and all their induced odd cycles must be $C_k$. In this paper, we present a polynomial-time algorithm that solves the 3-coloring problem for any graph in $\mathcal{G}_{10,7}$.

math.CO↗
arXiv2023

Complexity of $C_k$-coloring in hereditary classes of graphs

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in the complexity of the problem $H$-{\sc Coloring}, which asks for the existence of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc Coloring} of $F$-free graphs, where $F$ is a fixed graph and $H$ is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-{\sc Coloring} of $P_t$-free graphs. We show that for every odd $k \geq 5$ the $C_k$-{\sc Coloring} problem, even in the list variant, can be solved in polynomial time in $P_9$-free graphs. The algorithm extends for the case of list version of $C_k$-{\sc Coloring}, where $k$ is an even number of length at least 10. On the other hand, we prove that if some component of $F$ is not a subgraph of a subdividecd claw, then the following problems are NP-complete in $F$-free graphs: a)extension version of $C_k$-{\sc Coloring} for every odd $k \geq 5$, b) list version of $C_k$-{\sc Coloring} for every even $k \geq 6$.

cs.DS↗
arXiv2020

List-three-coloring $ P_t $-free graphs with no induced 1-subdivision of $ K_{1,s} $

Let $s$ and $t$ be positive integers. We use $P_t$ to denote the path with $t$ vertices and $K_{1,s}$ to denote the complete bipartite graph with parts of size $1$ and $s$ respectively. The one-subdivision of $K_{1,s}$ is obtained by replacing every edge $\{u,v\}$ of $K_{1,s}$ by two edges $\{u,w\}$ and $\{v,w\}$ with a new vertex $w$. In this paper, we give a polynomial-time algorithm for the list-three-coloring problem restricted to the class of $P_t$-free graph with no induced 1-subdivision of $K_{1,s}$.

math.CO↗
arXiv2020

Better 3-coloring algorithms: excluding a triangle and a seven vertex path

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in [Combinatorica 38(4):779--801, 2018], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to $O(|V(G)|^2(|V(G)|+|E(G)|))$. Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in $\{P_t,C_{\ell}\}$-free graphs is finite.

math.CO↗
arXiv2018

Four-coloring $P_6$-free graphs. I. Extending an excellent precoloring

This is the first paper in a series whose goal is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the $4$-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time algorithm that determines if a special kind of precoloring of a $P_6$-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem.

math.CO↗
arXiv2018

List-three-coloring graphs with no induced $P_6+rP_3$

For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomial-time algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.

math.CO↗
arXiv2018

Four-coloring $P_6$-free graphs. II. Finding an excellent precoloring

This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the $4$-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time algorithm that starts with a $4$-precoloring of a graph with no induced six-vertex path, and outputs a polynomial-size collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem.

math.CO↗
arXiv2018

Obstructions for three-coloring and list three-coloring $H$-free graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the case when $H$ is connected. This solves a problem posed by Golovach et al. As a second result, we characterize all graphs $H$ for which there are only finitely many $H$-free minimal obstructions for list 3-colorability.

math.CO↗
arXiv2016

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.

math.CO↗
arXiv2015

Three-coloring graphs with no induced seven-vertex path II : using a triangle

In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial time algorithm for three-coloring triangle-free graphs with no induced seven-vertex path. Combined, our work shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time.

cs.DM↗