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Chính T. Hoàng

Publications and source records attributed to Chính T. Hoàng.

17 recordsLinked to original sources
arXiv2025

On the structure of ($4K_1$, $C_4$, $P_6$)-free graphs

Determining the complexity of colouring ($4K_1, C_4$)-free graph is a long open problem. Recently Penev showed that there is a polynomial-time algorithm to colour a ($4K_1, C_4, C_6$)-free graph. In this paper, we will prove that if $G$ is a ($4K_1, C_4, P_6$)-free graph that contains a $C_6$, then $G$ has bounded clique-width. To this purpose, we use a new method to bound the clique-width, that is of independent interest. As a consequence, there is a polynomial-time algorithm to colour ($4K_1, C_4, P_6$)-free graphs.

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arXiv2025

On the structure of perfectly divisible graphs

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. A graph $G$ is minimally non-perfectly divisible if $G$ is not perfectly divisible but each of its proper induced subgraph is. A set $C$ of vertices of $G$ is a clique cutset if $C$ induces a clique in $G$, and $G-C$ is disconnected. We prove that a $P_5$-free minimally non-perfectly divisible graph cannot contain a clique cutset. This result allows us to re-establish several theorems on the perfect divisibility of some classes of $P_5$-free graphs. We will show that recognizing perfectly divisible graphs is NP-hard.

math.CO↗
arXiv2025

On graphs without four-vertex induced subgraphs

Given a family F of graphs, a graph G is F-free if it does not contain any graph in F as an induced subgraph. The problem of determining the complexity of colouring (claw, 4K1)- free graphs is a well-known open problem. In this paper we solve the colouring problem for a subclass of (claw, 4K1)-free graphs. We design a polynomial-time algorithm to colour (claw, 4K1, bridge, C4-twin)-free graphs. This algorithm is derived from a structural theorem on (claw, 4K1, bridge, C4-twin)-free graphs.

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arXiv2025

On the structure of (dart, odd hole)-free graphs

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible.

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arXiv2023

Infinite families of $k$-vertex-critical ($P_5$, $C_5$)-free graphs

A graph is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes known constructions for $4$-vertex-critical $P_7$-free graphs and $5$-vertex-critical $P_5$-free graphs and is in contrast to the fact that there are only finitely many $5$-vertex-critical $(P_5,C_5)$-free graphs. In fact, our construction is actually even more well-structured, being $(2P_2,K_3+P_1,C_5)$-free.

math.CO↗
arXiv2022

A refinement on the structure of vertex-critical ($P_5$, gem)-free graphs

We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$.

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arXiv2022

Vertex-critical $(P_3+\ell P_1)$-free and vertex-critical (gem, co-gem)-free graphs

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ where $χ(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\ell P_1)$-free graphs for all $k$ and all $\ell$. Together with previous results, the only graphs $H$ for which it is unknown if there are an infinite number of $k$-vertex-critical $H$-free graphs is $H=(P_4+\ell P_1)$ for all $\ell\ge 1$. We consider a restriction on the smallest open case, and show that there are only finitely many $k$-vertex-critical (gem, co-gem)-free graphs for all $k$, where gem$=\overline{P_4+P_1}$. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of $C_5$. This characterization allows us to give the complete list of all $k$-vertex-critical (gem, co-gem)-free graphs for all $k\le 16$

math.CO↗
arXiv2020

Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four

For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or $P_3 + P_1$. Our results imply the existence of new polynomial-time certifying algorithms for deciding the $k$-colorability of $(P_2+\ell P_1)$-free graphs for fixed $k$.

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arXiv2018

Classes of graphs with e-positive chromatic symmetric function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are e-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are e-positive. We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all e-positive, and conjecture that a graph is strongly e-positive if and only if it is (claw, net)-free.

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arXiv2017

On the structure of (banner, odd hole)-free graphs

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomial-time algorithms to find, for such a graph, a minimum coloring and largest stable set. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (banner, odd hole)-free graphs are perfect-divisible. %

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arXiv2017

Chromatic symmetric functions and H-free graphs

Two celebrated conjectures in chromatic symmetric function theory concern the positivity chromatics symmetric functions of claw-free graphs. Here we extend the claw-free idea to general graphs and consider the e-positivity question for H-free graphs where H = {claw, F} and H={claw, F, co-F}, where F is a four-vertex graph. We settle the question for all cases except H={claw, co-diamond}, and we provide some partial results in that case.

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arXiv2015

A Coloring Algorithm for $4K_1$-free line graphs

Let $L$ be a set of graphs. $Free$($L$) is the set of graphs that do not contain any graph in $L$ as an induced subgraph. It is known that if $L$ is a set of four-vertex graphs, then the complexity of the coloring problem for $Free$($L$) is known with three exceptions: $L $= {claw, $4K_1$}, $L$ = {claw, $4K_1$, co-diamond}, and $L$ = {$C_4$, $4K_1$}. In this paper, we study the coloring problem for $Free$(claw, $4K_1$). We solve the coloring problem for a subclass of $Free$(claw, $4K_1$) which contains the class of $4K_1$-free line graphs. Our result implies the chromatic index of a graph with no matching of size four can be computed in polynomial time.

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arXiv2014

Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: $\llcorner$,$\ulcorner$, $\urcorner$, $\lrcorner$, and we consider zero bend paths (i.e., | and $-$) to be degenerate $\llcorner$s. These graphs, called $B_1$-EPG graphs, were first introduced by Golumbic et al (2009). We consider the natural subclasses of $B_1$-EPG formed by the subsets of the four single bend shapes (i.e., {$\llcorner$}, {$\llcorner$,$\ulcorner$}, {$\llcorner$,$\urcorner$}, and {$\llcorner$,$\ulcorner$,$\urcorner$}) and we denote the classes by [$\llcorner$], [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$], and [$\llcorner$,$\ulcorner$,$\urcorner$] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [$\llcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$,$\urcorner$] $\subsetneq$ $B_1$-EPG; also, [$\llcorner$,$\ulcorner$] is incomparable with [$\llcorner$,$\urcorner$]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split $\cap$ [$\llcorner$].

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arXiv2014

On color-critical ($P_{5},\overline{P}_5$)-free graphs

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical $P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical $(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result implies the existence of a certifying algorithm for $k$-coloring $(P_5, \overline{P}_5)$-free graphs.

math.CO↗
arXiv2007

Deciding k-colourability of $P_5$-free graphs in polynomial time

The problem of computing the chromatic number of a $P_5$-free graph is known to be NP-hard. In contrast to this negative result, we show that determining whether or not a $P_5$-free graph admits a $k$-colouring, for each fixed number of colours $k$, can be done in polynomial time. If such a colouring exists, our algorithm produces it.

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