Songling Shan

Songling Shan
Illinois State University | ISU · Department of Mathematics

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68
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Publications (68)
Article
The toughness of a noncomplete graph G is the maximum real number t such that the ratio of |S| to the number of components of G−S is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant t0 such that every graph with toughness at least t0 is h...
Article
Let t>0 be a real number and let G be a graph. We say G is t-tough if for every cutset S of G, the ratio of |S| to the number of components of G−S is at least t. The Toughness Conjecture of Chvátal, stating that there exists a constant t0 such that every t0-tough graph with at least three vertices is hamiltonian, is still open in general. For any g...
Article
Let G $G$ be a simple graph with maximum degree Δ ( G ) ${\rm{\Delta }}(G)$. A subgraph H $H$ of G $G$ is overfull if ∣ E ( H ) ∣ > Δ ( G ) ⌊ ∣ V ( H ) ∣ ∕ 2 ⌋ . $| E(H)| \gt {\rm{\Delta }}(G)\lfloor | V(H)| \unicode{x02215}2\rfloor .$ Chetwynd and Hilton in 1986 conjectured that a graph G $G$ with Δ ( G ) > ∣ V ( G ) ∣ ∕ 3 ${\rm{\Delta }}(G)\gt |...
Article
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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Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and...
Article
Let m≥1 be an integer and G be a graph with m edges. We say that G has an antimagic orientation if G has an orientation D and a bijection τ:A(D)→{1,…,m} such that no two vertices in D have the same vertex-sum under τ, where the vertex-sum of a vertex v in D under τ is the sum of labels of all arcs entering v minus the sum of labels of all arcs leav...
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For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that every $2K_2$-free $3/2$-tough graph with at least three vertices has a $2$-factor, and the toughness bound $3...
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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...
Article
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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Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open...
Article
Full-text available
Let $G$ be a $t$-tough graph on $n\geqslant 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $t$ is between 1 and 2. In this paper, we show that if the degree...
Article
Full-text available
In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to Gallai’s question for many well-known classes of graphs such as split graphs, series–parallel graphs, and 2K2-free...
Preprint
Full-text available
An antimagic labeling of a digraph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\cdots,m\}$ such that all $n$ oriented vertex-sums are pairwise distinct, where the oriented vertex-sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph $...
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A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two. In this paper, we confirm the conjecture for $P_8$-free graphs by showing that there exists a c...
Preprint
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...
Preprint
Full-text available
The toughness of a noncomplete graph $G$ is the maximum real number $t$ such that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$ for every cutset $S$ of $G$, and the toughness of a complete graph is defined to be $\infty$. Determining the toughness for a given graph is NP-hard. Chv\'{a}tal's toughness conjecture, stating th...
Article
An antimagic labeling of a directed graph D with n vertices and m arcs is a bijection from the set of arcs of D to the integers { 1 , … , m } such that all n oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph has an an...
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Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $\tau:A(D)\rightarrow \{1,2,\ldots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $\tau$, where the vertex-sum of a vertex $v$ in $D$ under $\tau$ is the sum of labels of all...
Preprint
Full-text available
Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. The 1-factorization conjecture is a special c...
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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(...
Preprint
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Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K\"{u}hn and Osthus in 2016...
Preprint
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Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore's conditions in this direction, the problem was only studied when $t$ is between 1 and 2. In this paper, we show that if the degree sum o...
Preprint
Full-text available
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
Let m≥1 be an integer and G be a graph with m edges. We say that G has an antimagic orientation if G has an orientation D and a bijection τ:A(D)→{1,2,…,m} such that no two vertices in D have the same vertex-sum under τ, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs le...
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Full-text available
For an integer k , a k ‐tree is a tree with maximum degree at most k . More generally, if f is an integer‐valued function on vertices, an f ‐tree is a tree in which each vertex v has degree at most f ( v ) . Let c ( G ) denote the number of components of a graph G . We show that if G is a connected K 4 ‐minor‐free graph and c ( G − S ) ≤ ∑ v ∈ S (...
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...
Preprint
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...
Preprint
An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A...
Preprint
Full-text available
Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $\tau:A(D)\rightarrow \{1,2,\cdots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $\tau$, where the vertex-sum of a vertex $u$ in $D$ under $\tau$ is the sum of labels of all...
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Full-text available
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...
Article
The prism over a graph G is the product G□K2, i.e., the graph obtained by taking two copies of G and adding a perfect matching joining the two copies of each vertex by an edge. The graph G is called prism-hamiltonian if it has a hamiltonian prism. Jung showed that every 1-tough P4-free graph with at least three vertices is hamiltonian. In this pape...
Preprint
Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that every digraph has a quasi-kernel. P. L. Erd\H{o}s and L. A. Sz\'ekely in 1976 conjectured that if every vertex...
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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...
Article
A graph is called 2 K 2‐free if it does not contain two independent edges as an induced subgraph. Broersma, Patel, and Pyatkin showed that every 25‐tough 2 K 2‐free graph with at least three vertices is Hamiltonian. In this paper, we improve the required toughness in this result from 25 to 3.
Preprint
In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to Gallai's question for many well-known classes of graphs such as split graphs, series parallel graphs, and $2K_2$-f...
Article
Full-text available
Let G be a simple graph, and let ∆(G) andχ′(G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ′(G) = ∆(G) or χ′(G) = ∆(G)+1. We say G is ∆-critical ifχ′(G) = ∆(G)+1 andχ′(H)< χ′(G) for every proper subgraph H of G. In 1968, Vizing conjectured that if G is a ∆-critical graph, then G has a 2-factor. Let G be an...
Preprint
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-toug...
Preprint
The \emph{prism} over a graph $G$ is the product $G \Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vert...
Preprint
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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An (r - 1; 1)-coloring of an r-regular graph G is an edge coloring (with arbitrarily many colors) such that each vertex is incident to r - 1 edges of one color and 1 edge of a different color. In this paper, we completely characterize all 4-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a (3; 1)-coloring....
Article
Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a...
Article
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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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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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A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Broersma, Patel, and Pyatkin showed that every 25-tough $2K_2$-free graph with at least three vertices is hamiltonian. In this paper, we improve the required toughness in this result from 25 to 3.
Article
Full-text available
An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An...
Article
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...
Article
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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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In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs....
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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...
Article
An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor o...
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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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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 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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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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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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It has been known that determining the exact value of vertex distinguishing edge index χs′(G) of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let nd(G) denote the number of vertices of degree d in G, and let χ′es(G)χ′es(G) be the equi...
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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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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 C m,n be the join graph of C m (a cycle of length m) and n isolated vertices. In this paper, we first show that the genus and nonorientable genus of C m,n equal those of K m,n . Then we show that the complete bipartite graph K m,n has a strong orientable genus embedding if m⩾2 and n⩾2 and has a strong nonorientable genus embedding if m⩾3 and n⩾...

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