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Chinh T. Hoang

Publications and source records attributed to Chinh T. Hoang.

9 recordsLinked to original sources
arXiv2026

Structural description of (bull, house)-free graphs

The bull is a graph consisting of a triangle and two pendant edges. The P_5 is the chordless path on five vertices. The house is the complement of a P_5. 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 k-critical P_5-free graphs and bull-free graphs are infinite for large enough k. We give a structural description of (bull, house)-free graphs and also (bull, P_5)-free graphs. Using these structural properties we prove that for any fixed k, the number of k-critical (bull, P_5)-free graphs is finite. This improves on a result of Huang, Li and Xia (Critical (P_5, bull)-free graphs, Discrete Applied Mathematics 334 (2023) 15-25). A graph G is perfectly divisible if for each induced subgraph H of G with at least one edge, V(H) can be partitioned into two sets V_1, V_2 such that every largest clique of H contains a vertex in V_i for i = 1,2. Chudnovsky and Sivaraman proved that (P_5, bull)-free graphs are perfectly divisible (Perfect divisibility and 2-divisibility, Journal of Graph Theory 90 (2019) 54-60). Our structural result allows us to give a short proof of this theorem.

math.CO↗
arXiv2023

A Closure Lemma for tough graphs and Hamiltonian degree conditions

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma proved by Bondy and Chvátal states that a graph $G$ is Hamiltonian if and only if its closure $G^*$ is. This lemma can be used to prove several classical results in Hamiltonian graph theory. We prove a version of the Closure Lemma for tough graphs. A graph $G$ is $t$-tough if for any set $S$ of vertices of $G$, the number of components of $G-S$ is at most $t |S|$. A Hamiltonian graph must necessarily be 1-tough. Conversely, Chvátal conjectured that there exists a constant $t$ such that every $t$-tough graph is Hamiltonian. The {\it $t$-closure} of a graph $G$ is the graph $G^{t*}$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n-t$. We prove that, for $t\geq 2$, a $\frac{3t-1}{2}$-tough graph $G$ is Hamiltonian if and only if its $t$-closure $G^{t*}$ is. Hoàng conjectured the following: Let $G$ be a graph with degree sequence $d_1 \leq d_2 \leq \ldots \leq d_n$; then $G$ is Hamiltonian if $G$ is $t$-tough and, $\forall i <\frac{n}{2},\mbox{ if } d_i\leq i \mbox{ then } d_{n-i+t}\geq n-i$. This conjecture is analogous to the well known theorem of Chvátal on Hamiltonian ideals. Hoàng proved the conjecture for $t \leq 3$. Using the closure lemma for tough graphs, we prove the conjecture for $t = 4$.

math.CO↗
arXiv2019

The intersection of two vertex coloring problems

A hole is an induced cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two problems are unresolved: the complexity of coloring even hole-free graphs, and the complexity of coloring {4K1, C4}-free graphs. The intersection of these two problems is the problem of coloring {4K1, C4, C6}-free graphs. In this paper we present partial results on this problem.

cs.DM↗
arXiv2017

On the structure of (pan, even hole)-free graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our $O(nm)$-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our $O(n^{2.5}+nm)$-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.

cs.DM↗
arXiv2009

Finding a sun in building-free graphs

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete. We show that whether a building-free graph contains a sun can be decided in O(min$\{m{n^3}, m^{1.5}n^2\}$) time and, if a sun exists, it can be found in the same time bound. The class of building-free graphs contains many interesting classes of perfect graphs such as Meyniel graphs which, in turn, contains classes such as hhd-free graphs, i-triangulated graphs, and parity graphs. Moreover, there are imperfect graphs that are building-free. The class of building-free graphs generalizes several classes of graphs for which an efficient test for the presence of a sun is known. We also present a vertex elimination scheme for the class of (building, gem)-free graphs. The class of (building, gem)-free graphs is a generalization of the class of distance hereditary graphs and a restriction of the class of (building, sun)-free graphs.

cs.DM↗
arXiv2009

On graphs without a C4 or a diamond

We consider the class of (C4, diamond)-free graphs; graphs in this class do not contain a C4 or a diamond as an induced subgraph. We provide an efficient recognition algorithm for this class. We count the number of maximal cliques in a (C4, diamond)-free graph and the number of n-vertex, labeled (C4, diamond)-free graphs. We also give an efficient algorithm for finding a largest clique in the more general class of (house, diamond)-free graphs.

cs.DM↗
arXiv2009

On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two

In an article [3] published recently in this journal, it was shown that when k >= 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to the issue of solvability in polynomial time, we show that the problem is at least as hard as graph automorphism but no harder than graph isomorphism.

cs.CC↗
arXiv2008

On the complexity of finding a sun in a graph

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs, have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun in NP-complete.

cs.DM↗