Guantao Chen's research while affiliated with Georgia State University and other places

Publications (174)

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Let G be a simple graph, and let n, Δ(G) and χ′(G) be the order, the maximum degree and the chromatic index of G, respectively. We call G overfull if |E(G)|/⌊n/2⌋>Δ(G), and critical if χ′(H)<χ′(G) for every proper subgraph H of G. Clearly, if G is overfull then χ′(G)=Δ(G)+1 by Vizing's Theorem. The core of G, denoted by GΔ, is the subgraph of G ind...
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Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\it critical} if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi'(G) = \D...
Article
For a simple graph G, denote by n, Δ(G), and χ′(G) its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if χ′(G)=Δ(G)+1 and χ′(H)<χ′(G) for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let G⁎ be obtained from G by splitting one vertex of G into two vertices. Hi...
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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, $\operatorname{la}(G) \ge \lceil\Delta(G)/2\rceil$ for a graph $G$ with maximum degree $\Delta(G)$. On the other hand, the Linear...
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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 distanc...
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Let G $G$ be a simple graph with maximum degree Δ(G) ${\rm{\Delta }}(G)$ and chromatic index χ′(G) $\chi ^{\prime} (G)$. A classical result of Vizing shows that either χ′(G)=Δ(G) $\chi ^{\prime} (G)={\rm{\Delta }}(G)$ or χ′(G)=Δ(G)+1 $\chi ^{\prime} (G)={\rm{\Delta }}(G)+1$. A simple graph G $G$ is called edge‐Δ ${\rm{\Delta }}$‐critical if G $G$ i...
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We call a graph G $G$ a k $k$‐threshold graph if there are k $k$ distinct real numbers θ 1 , θ 2 , … , θ k ${\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{k}$ and a mapping r : V ( G ) → R $r:V(G)\to {\mathbb{R}}$ such that for any two vertices u , v ∈ V ( G ) $u,v\in V(G)$, we have that u v ∈ E ( G ) $uv\in E(G)$ if and only if there are odd numbe...
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Let G=(V(G),E(G)) be a multigraph with maximum degree Δ(G), chromatic index χ′(G), and total chromatic number χ″(G). The total coloring conjecture proposed by Behzad and Vizing, independently, states that χ″(G)≤Δ(G)+μ(G)+1 for a multigraph G, where μ(G) is the multiplicity of G. Moreover, Goldberg conjectured that χ″(G)=χ′(G) if χ′(G)≥Δ(G)+3 and no...
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Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq \Delta(G)+\mu(G) +1$ for a multigraph $G$, where $\mu(G)$ is the multiplicity of $G$. Moreover, Goldberg conje...
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An edge cut C of a graph G is tight if |C∩M|=1 for every perfect matching M of G. Barrier cuts and 2-separation cuts are called ELP-cuts, which are two important types of tight cuts in matching covered graphs. Edmonds, Lovász and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. C...
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A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a s...
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A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a s...
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A graph is a linear forest if each of its components is a path. Given a graph G with maximum degree Δ(G), motivated by the famous linear arboricity conjecture and Lovász's classic result on partitioning the edge set of a graph into paths, we call a partition F:=F1|⋯|Fk of the edge set of G an exact linear forest partition if each Fi induces a linea...
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For a given graph H, the Ramsey number r(H) is the minimum N such that any 2-edge-coloring of the complete graph KN yields a monochromatic copy of H. Given a positive integer n, a fanFn is a graph formed by n triangles that share one common vertex. We show that 9n∕2−5≤r(Fn)≤11n∕2+6 for any n. This improves previous best bounds r(Fn)≤6n of Lin and L...
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The multiplicity of the second‐largest eigenvalue of the adjacency matrix A ( G ) of a connected graph G, denoted by m ( λ 2 , G ), is the number of times of the second‐largest eigenvalue of A ( G ) appears. In 2019, Jiang, Tidor, Yao, Zhang, and Zhao gave an upper bound on m ( λ 2 , G ) for graphs G with bounded degrees, and applied it to solve a...
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Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$\Delta$-critical} if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ of $G$; and $G$ is \emph{overfull} if $|E(G)|>\Delta \lfloor |V(...
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For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$ of $G$. Define $G$ to be overfull if $|E(G)|>\Delta(G) \lfloor n/2 \rfloor$. Clearly, overfull graphs are class 2 and...
Article
In the study of graph edge coloring for simple graphs, a graph G is called Δ-critical if Δ(G)=Δ, χ′(G)=Δ(G)+1 and χ′(H)<χ′(G) for every proper subgraph H of G. In this paper, we prove a new adjacency result of critical graphs which allows us to control the degree of vertices with distance four. Combining this result with a previous theorem proved b...
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A graph is said to be ISK 4 ‐free if it does not contain any subdivision of K 4 as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ISK 4 ‐free graph is 4‐colorable. In this paper, we show that this conjecture is true for the class of { ISK 4 , diamond, bowtie}‐free graphs, where a diamond is the graph obtained from K 4 by...
Preprint
Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi'(G) = \Delta(G)+1$. The...
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For a given graph $H$, the Ramsey number $r(H)$ is the minimum $N$ such that any 2-edge-coloring of the complete graph $K_N$ yields a monochromatic copy of $H$. Given a positive integer $n$, let $nK_3$, $F_n$ and $B_n$ be three graphs formed by $n$ triangles that share zero, one, and two common vertices, respectively. Burr, Erd\H{o}s and Spencer in...
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Let $G$ be a simple graph with maximum degree $\Delta$. A classic result of Vizing shows that $\chi'(G)$, the chromatic index of $G$, is either $\Delta$ or $\Delta+1$. We say $G$ is of \emph{Class 1} if $\chi'(G)=\Delta$, and is of \emph{Class 2} otherwise. A graph $G$ is \emph{$\Delta$-critical} if $\chi'(G)=\Delta+1$ and $\chi'(H)<\Delta+1$ for e...
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Let G be a simple graph, and let χ′(G) and Δ(G) denote the chromatic index and the maximum degree of G, respectively. A graph G is a critical class two graph if χ′(G)=Δ(G)+1 and χ′(H)≤Δ(G) for every proper subgraph H of G. Let d‾(G) denote the average degree of G, i.e., d‾(G)=2|E(G)|/|V(G)|. Vizing in 1968 conjectured that d‾(G)≥Δ(G)−1+3/n if G is...
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Let $G$ be a simple graph with maximum degree $\Delta$. We call $G$ \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. A classic result of Vizing shows that $\chi'(G)$, the chromatic index of $G$, is either $\Delta$ or $\Delta+1...
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An edge cut $C$ of a graph $G$ is {\it tight} if $|C \cap M|=1$ for every perfect matching $M$ of $G$.~Barrier cuts and 2-separation cuts are called {\it ELP-cuts}, which are two important types of tight cuts in matching covered graphs.~Edmonds, Lov\'asz and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it als...
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In 2012, Lévêque, Maffray and Trotignon conjectured that if a graph does not contain an induced subdivision of \(K_4\), then it is 4-colorable. Recently, Le showed that every such graph is 24-colorable. In this paper, we improve the upper bound to 8.
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A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known t...
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For two graphs A and B, a graph G is called \(\{A,B\}\)-free if G contains neither A nor B as an induced subgraph. Let \(P_{n}\) denote the path of order n. For nonnegative integers k, \(\ell \) and m, let \(N_{k,\ell ,m}\) be the graph obtained from \(K_{3}\) and three vertex-disjoint paths \(P_{k+1}\), \(P_{\ell +1}\), \(P_{m+1}\) by identifying...
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The classical Ore’s Theorem states that every graph G of order n≥3 with σ2(G)≥n is hamiltonian, where σ2(G)=min{dG(x)+dG(y):x,y∈V(G),x≠y,xy∉E(G)}. Recently, Ferrara, Jacobson and Powell (Discrete Math. 312 (2012), 459–461) extended the Moon–Moser Theorem and characterized the non-hamiltonian balanced bipartite graphs H of order 2n≥4 with partite se...
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Let G be a multigraph with maximum degree Δ and chromatic index χ′. If G is bipartite then χ′=Δ. Otherwise, by a theorem of Goldberg, χ′≤Δ+1+⌊(Δ−2)/(go−1)⌋, where go denotes the odd girth of G. Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if χ′=Δ+1+⌊(Δ−2)/(go−1)⌋ then G contains as a subgraph a ring graph R with the same ch...
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Let $G=(V,E)$ be a multigraph. The {\em cover index} $\xi(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. Let $\delta(G)$ be the minimum degree of $G$ and let $\Phi(G)$ be the {\em co-density} of $G$, defined by \[\Phi(G)=\min...
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Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In th...
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Graph edge coloring has a rich theory, many applications and beautiful conjectures, and it is studied not only by mathematicians, but also by computer scientists. In this survey, written for the non-expert, we shall describe some main results and techniques and state some of the many popular conjectures in the theory. Besides known results a new ba...
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Carvalho, Lucchesi, and Murty [J. Combin. Theory Ser. B, 92 (2004), pp. 319-324, Theorem 3.5] presented a proof of a theorem of Reed and Wakabayashi that a brick G is nonsolid if and only if there exist two vertex-disjoint odd cycles C 1 and C 2 such that G - V (C 1 C 2 ) has a perfect matching. Consequently, every brick with no two vertex-disjoint...
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A cycle of order k is called a k-cycle. A non-induced cycle is called a chorded cycle. Let n be an integer with n≥4. Then a graph G of order n is chorded pancyclic if G contains a chorded k-cycle for every integer k with 4≤k≤n. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order n satisfying degGu+degGv...
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Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A classic result of Vizing indicates that either $\chi'(G )=\Delta(G)$ or $\chi'(G )=\Delta(G)+1$. The graph $G$ is called $\Delta$-critical if $G$ is connected, $\chi'(G )=\Delta(G)+1$ and for any $e\in E(G)$, $\chi'(G-e)=\Delta(G)$. Let $G$ be an $n$-vertex...
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Let $G=(V,E)$ be a finite undirected graph and let $R, S\subseteq V$. If $P$ and $Q$ are disjoint oriented paths in $G$, with $P$ connecting $r_1\in R$ to $r_2\in R$ and with $Q$ connecting $s_1\in S$ to $s_2\in S$, we define $\pi(P, Q) = r_1\otimes s_1 - r_1\otimes s_2 - r_2\otimes s_1 + r_2\otimes s_2\in \mathbb{Z}\langle R\rangle\otimes \mathbb{...
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Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k \ge \Delta +1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $\Delta$ denotes the maximum degree of $G$. Tashkinov tree met...
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A Halin graph is constructed from a plane embedding of a tree with no vertices of degree 2 by adding a cycle through its leaves in the natural order determined by the embedding. Halin graphs satisfy interesting properties. However, to our knowledge, there are no results giving a positive answer for “spanning Halin subgraph problem” (i.e., which gra...
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Given a graph $G$, denote by $\Delta$ and $\chi^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maxim...
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Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is e...
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Vizing conjectured that every edge chromatic critical graph contains a 2-factor. Believing that stronger properties hold for this class of graphs, Luo and Zhao (2013) showed that every edge chromatic critical graph of order n with maximum degree at least 6n7 is Hamiltonian. Furthermore, Luo et al. (2016) proved that every edge chromatic critical gr...
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Wu et al. (Discret Math 313:2696–2701, 2013) conjectured that the vertex set of any simple graph G can be equitably partitioned into m subsets so that each subset induces a forest, where Δ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \u...
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In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A g...
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Let G be an n-vertex simple graph, and let and denote the maximum degree and chromatic index of G, respectively. Vizing proved that or . Define G to be Δ-critical if and for every proper subgraph H of G. In 1965, Vizing conjectured that if G is an n-vertex Δ-critical graph, then G has a 2-factor. Luo and Zhao showed if G is an n-vertex Δ-critical g...
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A Halin graph is a plane graph constructed from a planar drawing of a tree by connecting all leaves of the tree with a cycle which passes around the boundary of the graph. The tree must have four or more vertices and no vertices of degree two. Halin graphs have many nice properties such as being Hamiltonian and remaining Hamiltonian after any singl...
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A graph is called 2K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2K_2$$\end{document}-free if it does not contain two independent edges as an induced subgraph. Gao...
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Given a graph $G$ possibly with multiple edges but no loops, denote by $\Delta$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, $\chi'$ the {\it chromatic index} and $\chi_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $\Delta\le \chi_f' \le \chi' \le \Delta + \mu$, where the upper bound is a classic result...
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This paper aims to review state-of-the-art Bayesian-inference-based methods applied to functional magnetic resonance imaging (fMRI) data. Particularly, we focus on one specific long-standing challenge in the computational modeling of fMRI datasets: how to effectively explore typical functional interactions from fMRI time series and the correspondin...
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Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in:...
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When we study forbidden subgraph conditions guaranteeing graphs to have some properties, a claw (or $K_{1,3}$) frequently appears as one of forbidden subgraphs. Recently, Furuya and Tsuchiya compared two classes generated by different forbidden pairs containing a claw, and characterized one of such classes. In this paper, we give such characterizat...
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Hadwiger conjectured that every graph contains as a minor, where is the chromatic number of G. In 2005, Robertson made a weaker conjecture that for any graph G, there exists a graph H with the same degree sequence of G and containing as a minor, which was confirmed by Dvořák and Mohar recently. In this note, we give a short proof of Robertson's Con...
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Bollobás and Thomason showed that a multigraph of order n and size at least n + c (c ≥ 1) contains a cycle of length at most 2(⌊n/c⌋ + 1) ⌊log2 2c⌋. We show in this paper that a multigraph (with no loop) of order n and minimum degree at least 5 contains a chorded cycle (a cycle with a chord) of length at most 300 log2 n. As an application of this r...
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A Halin graph is a simple plane graph consisting of a tree without degree 2 vertices and a cycle induced by the leaves of the tree. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this paper, we construct an infinite family of counterexamples to the conjecture.
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A spanning tree with no vertices of degree 2 is called a Homeomorphically irreducible spanning tree\,(HIST). Based on a HIST embedded in the plane, a Halin graph is formed by connecting the leaves of the tree into a cycle following the cyclic order determined by the embedding. Both of the determination problems of whether a graph contains a HIST or...
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The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if $C$ is a hamiltonian cycle of a graph $G$, then the square of $C$ is called a hamiltonian square of $G$. In this paper, we characterize all possible forbidden pairs, which implies the containment of a ham...
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A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a s...
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A tree with atmost m leaves is called an m-ended tree. Kyaw proved that every connected K 1,4-free graph with σ 4(G) ⩽ n−1 contains a spanning 3-ended tree. In this paper we obtain a result for k-connected K 1,4-free graphs with k ⩽ 2. Let G be a k-connected K 1,4-free graph of order n with k ⩽ 2. If σ k+3(G) ⩽ n+2k −2, then G contains a spanning 3...
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A set F of edges in a digraph D is called a directed cut if there exists a nontrivial partition (X,Y) of V(D) such that F consists of all directed edges from X to Y. Let Λ(D) denote the maximum size of a directed cut of D, and let D(1,1) be the set of all digraphs D such that d+(v)=1 or d-(v)=1 for any vertex v in D. We show that Λ(D)>= 38(|E(D)|-1...
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In 1968, Vizing conjectured that, if $G$ is an $n$-vertex edge chromatic critical graph with $\chi'(G)=\Delta(G)+1$, then $G$ contains a 2-factor. In this paper, we verify this conjecture for $n\le 2\Delta(G)$.
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Chvátal and Erdős proved a well-known result that the graph $$G$$G with connectivity $$\kappa (G)$$κ(G) not less than its independence number $$\alpha (G)$$α(G) [$$\alpha (G)+1$$α(G)+1, $$\alpha (G)-1$$α(G)-1, respectively] is Hamiltonian (traceable, Hamiltonian-connected, respectively). In this paper, we strengthen the Chvátal–Erdős theorem to the...
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Chlamydia trachomatis (CT) and Neisseria gonorrhoeae (GC) are the agents of two common, sexually transmitted diseases afflicting women in the United States (http://www.cdc.gov). We designed a novel web-based application that offers simple recommendations to help optimize medical outcomes with CT and GC prevention and control programs. This applicat...
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We show that if G is a graph such that every edge is in at least two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree). This result was originally asked in a question format by Albertson, Berman, Hutchinson, and Thomassen in 1979, and then conjectured to be true by Archdeacon in 200...
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The well-known Chvátal-Erdős theorem states that every graph G of order at least three with α(G)≤κ(G) has a Hamiltonian cycle, where α(G) and κ(G) are the independence number and the connectivity of G, respectively. D. J. Oberly and D. P. Sumner [J. Graph Theory 3, 351–356 (1979; Zbl 0424.05036)] have proved that every connected, locally connected...
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Identifying Ca(2+) -binding sites in proteins is the first step toward understanding the molecular basis of diseases related to Ca(2+) -binding proteins. Currently, these sites are identified in structures either through X-ray crystallography or NMR analysis. However, Ca(2+) -binding sites are not always visible in X-ray structures due to flexibili...
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A path in a graph is called extendable if it is a proper subpath of another path. A graph is locally connected if every neighborhood induces a connected subgraph. We show that, for each graph G of order n, there exists a threshold number s such that every path of order smaller than s is extendable and there exists a non-extendable path of order t f...
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Chlamydia trachomatis (CT) and Neisseria gonorrhoeae (GC) are two common sexually transmitted diseases (STDs) in the United States. Annual screening for CT for all sexually active women aged 25 years and younger, all pregnant women, women with history of STDs, or women older than 25 years who are at increased risk of infection (e.g., women who have...
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For an integer k with k≥2 and a pair of connected graphs F1 and F2 of order at least three, we say that {F1,F2} is a k-forbidden pair if every k-connected {F1,F2}-free graph, except possibly for a finite number of exceptions, is Hamiltonian. If no exception arises, {F1,F2} is said to be a strong k-forbidden pair. The 2-forbidden pairs and the stron...
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A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more...
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Let G be a graph of order n ≥ 3. A subgraph H of G is called a square hamiltonian cycle if it consists of a hamiltonian cycle v1v2 ...v nv1 and chords vivi+2 for all i =1 , 2 ,...,n (where vn+j = vj for 1 ≤ j ≤ n). Clearly, a square hamiltonian cycle contains all possible 2-regular graphs of order n. Fan and Kiestead showed that, for every positive...
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Chlamydia trachomatis (CT) and Neisseria gonorrhoeae (GC) are two common sexually transmitted diseases among women in the United States. Publicly funded programs usually do not have enough money to screen and treat all patients. Therefore, the authors propose a new resource allocation model to assist clinical managers to make decisions on identifyi...
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Let G be a k-connected graph of order n, α: = α(G) the independence number of G, and c(G) the circumference of G. Chvátal and Erdős proved that if α⩽kthen G is hamiltonian. For α⩾k⩾2, Fouquet and Jolivet in 1978 made the conjecture that c(G)⩾k(n+ α − k)/α. Fournier proved that the conjecture is true for α⩽k+ 2 or k = 2 in two different papers. Mano...
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The toughness of a non-complete graph G is the minimum value of |S| ω(G-S) among all separating vertex sets S⊂V(G), where ω(G-S)⩾2 is the number of components of G-S. It is well-known that every 3-connected planar graph has toughness greater than 1/2. Related to this property, every 3-connected planar graph has many good substructures, such as a sp...
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The complete area coverage problem in Wireless Sensor Networks (WSNs) has been extensively studied in the literature. However, many applications do not require complete coverage all the time. For such applications, one effective method to save energy and prolong network lifetime is to partially cover the area. This method for prolonging network lif...
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It is well known that if G is a multigraph then χ′(G)≥χ′*(G):=max {Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′*(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max {2|E(G[U])|/(|U|−1):U⊆V(G),|U|≥3, |U| is odd}. The conjecture that χ′(G)≤max {Δ(G)+1,⌈Γ(G)⌉} was made independently by Goldberg (Discret....
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Given a family of graphs ℱ, a graph G is ℱ-saturated if no element of ℱ is a subgraph of G, but for any edge e in G ¯, some element of ℱ is a subgraph of G+e. Let sat(n,ℱ) denote the minimum number of edges in an ℱ-saturated graph of order n. For graphs G, H 1 ,⋯,H k , we write that G→(H 1 ,⋯,H k ) if every k-coloring of E(G) contains a monochromat...