arXiv subjects

Songling Shan

Publications and source records attributed to Songling Shan.

At least 55 records · Page 3Linked to original sources

arXiv2022

Precoloring extension of Vizing's Theorem for multigraphs

Let $G$ be a graph with maximum degree $Δ(G)$ and maximum multiplicity $μ(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $Δ(G)+μ(G)$. 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 distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, Δ(G)+μ(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge coloring of $G$. Girão and Kang verified this conjecture for distance-$9$ matchings. In this paper, we improve the required distance from $9$ to $3$ for multigraphs $G$ with $μ(G) \ge 2$.

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arXiv2022

A note on hamiltonian cycles in $4$-tough $(P_2\cup kP_1)$-free graphs

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 Chvátal, 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 in general. For any given integer $k\ge 1$, a graph $G$ is $(P_2\cup kP_1)$ free if $G$ does not contain the disjoint union of $P_2$ and $k$ isolated vertices as an induced subgraph. In this note, we show that every 4-tough and $2k$-connected $(P_2\cup kP_1)$-free graph with at least three vertices is hamiltonian. This result in some sense is an "extension" of the classical Chvátal-Erdős Theorem that every $\max\{2,k\}$-connected $(k+1)P_1$-free graph on at least three vertices is hamiltonian.

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arXiv2022

An Ore-type condition for hamiltonicity in tough graphs

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-type hamiltonicity conditions, the problem was only studied when $t$ is between 1 and 2. In this paper, we show that if the degree sum of any two nonadjacent vertices of $G$ is greater than $\frac{2n}{t+1}+t-2$, then $G$ is hamiltonian.

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arXiv2021

Antimagic Orientation of Forests

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 $G$ admits an antimagic orientation if $G$ has an orientation $D$ such that $D$ has an antimagic labeling. Hefetz, M{ü}tze and Schwartz conjectured every connected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that any forest obtained from a given forest with at most one isolated vertex by subdividing each edge at least once admits an antimagic orientation.

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arXiv2021

Erdős-Gyárfás Conjecture for $P_8$-free graphs

A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erdős and Gyárfás 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 cycle of length four or eight in every $P_8$-free graph with minimum degree at least three.

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arXiv2021

The Core Conjecture of Hilton and Zhao II: a Proof

A simple graph $G$ with maximum degree $Δ$ is overfull if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. Clearly, the chromatic index of $G$ equals $Δ+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case $Δ=3$ in 2003, and Cranston and Rabern proved the next case $Δ=4$ in 2019. In this paper, we give a proof of this conjecture for all $Δ\ge 4$.

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arXiv2021

Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs

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. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.

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arXiv2021

Edge coloring graphs with large minimum degree

Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Δ(G)>|V(G)|/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. The 1-factorization conjecture is a special case of this overfull conjecture, which states that for even $n$, every regular $n$-vertex graph with degree at least about $n/2$ has a 1-factorization and was confirmed for large graphs in 2014. Supporting the overfull conjecture as well as generalizing the 1-factorization conjecture in an asymptotic way, in this paper, we show that for any given $0<\varepsilon <1$, there exists a positive integer $n_0$ such that the following statement holds: if $G$ is a graph on $2n\ge n_0$ vertices with minimum degree at least $(1+\varepsilon)n$, then $G$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph.

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arXiv2021

Antimagic orientation of subdivided caterpillars

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 $τ:A(D)\rightarrow \{1,2,\ldots,m\}$ 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 leaving $v$. Hefetz, Mütze and Schwartz [J. Graph Theory, 64: 219-232, 2010] conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as regular graphs, graphs with minimum degree at least 33, bipartite graphs with no vertex of degree zero or two, and trees including caterpillars and complete $k$-ary trees. We prove that every subdivided caterpillar admits an antimagic orientation, where a subdivided caterpillar is a subdivision of a caterpillar $T$ such that the edges of $T$ that are not on the central path of $T$ are subdivided the same number of times.

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arXiv2021

The overfullness of graphs with small minimum degree and large maximum degree

Given a simple graph $G$, denote by $Δ(G)$, $δ(G)$, and $χ'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ(G)+1$ and $χ'(H)\le Δ(G)$ for every proper subgraph $H$ of $G$; and $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $χ'(G) = Δ(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $Δ$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $Δ(G) > |V(G)|/3$. In this paper, we show that any $Δ$-critical graph $G$ is overfull if $Δ(G) - 7δ(G)/4\ge(3|V(G)|-17)/4$.

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arXiv2021

Chromatic index of dense quasirandom graphs

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arXiv2021

An improvement to the vertex-splitting conjecture

For a simple graph $G$, denote by $n$, $Δ(G)$, and $χ'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $χ'(G-e)<Δ(G)+1$ for every edge $e$ of $G$. Define $G$ to be overfull if $|E(G)|>Δ(G) \lfloor n/2 \rfloor$. Clearly, overfull graphs are class 2 and any graph obtained from a regular graph of even order by splitting a vertex is overfull. Let $G$ be an $n$-vertex connected regular class 1 graph with $Δ(G) >n/3$. Hilton and Zhao in 1997 conjectured that if $G^*$ is obtained from $G$ by splitting one vertex of $G$ into two vertices, then $G^*$ is edge-chromatic critical, and they verified the conjecture for graphs $G$ with $Δ(G)\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. The graph $G^*$ is easily verified to be overfull, and so the hardness of the conjecture lies in showing that the deletion of every of its edge decreases the chromatic index. Except in 2002, Song showed that the conjecture is true for a special class of graphs $G$ with $Δ(G)\ge \frac{n}{2}$, no other progress on this conjecture had been made. In this paper, we confirm the conjecture for graphs $G$ with $Δ(G) \ge 0.75n$.

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arXiv2020

Overfullness of critical class 2 graphs with a small core degree

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 $|E(G)|/\lfloor n/2\rfloor > Δ(G)$, and critical if $χ'(H) < χ'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $χ'(G) = Δ(G)+1$. The core of $G$, denoted by $G_Δ$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $Δ(G) \ge 4$, if the maximum degree of $G_Δ$ is at most two, then $G$ is overfull, which in turn gives $Δ(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_Δ$ is at most two and $Δ(G) > n/2 +1$, then $G$ is overfull.

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arXiv2020

$Δ$-critical graphs with a vertex of degree 2

Let $G$ be a simple graph with maximum degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. We say $G$ is of \emph{Class 1} if $χ'(G)=Δ$, and is of \emph{Class 2} otherwise. A graph $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ+1$ and $χ'(H)<Δ+1$ for every proper subgraph $H$ of $G$, and is \emph{overfull} if $|E(G)|>Δ\lfloor (|V(G)|-1)/2 \rfloor$. Clearly, overfull graphs are Class 2. Hilton and Zhao in 1997 conjectured that if $G$ is obtained from an $n$-vertex $Δ$-regular Class 1 graph with maximum degree greater than $n/3$ by splitting a vertex, then being overfull is the only reason for $G$ to be Class 2. This conjecture was only confirmed when $Δ\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. In this paper, we improve the bound on $Δ$ from $\frac{n}{2}(\sqrt{7}-1)$ to $0.75n$. Considering the structure of $Δ$-critical graphs with a vertex of degree 2, we also show that for an $n$-vertex $Δ$-critical graph with $Δ\ge \frac{3n}{4}$, if it contains a vertex of degree 2, then it is overfull. We actually obtain a more general form of this result, which partially supports the overfull conjecture of Chetwynd and Hilton from 1986, which states that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ>n/3$, then $G$ contains an overfull subgraph $H$ with $Δ(H)=Δ$. Our proof techniques are new and might shed some light on attacking both of the conjectures when $Δ$ is large.

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arXiv2020

Antimagic orientation of graphs with minimum degree at least 33

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 graph $G$ has an antimagic orientation if it has an orientation which admits an antimagic labeling. Hefetz, M{ü}tze, and Schwartz conjectured that every connected graph admits an antimagic orientation. In this paper, we show that every bipartite graph without both isolated and degree 2 vertices admits an antimagic orientation and every graph $G$ with $δ(G)\ge 33$ admits an antimagic orientation. Our proof relies on a newly developed structural property of bipartite graphs, which might be of independent interest.

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arXiv2020

Antimagic orientation of lobsters

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 $τ:A(D)\rightarrow \{1,2,\cdots,m\}$ 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 leaving $u$. Hefetz, Mütze and Schwartz [J. Graph Theory, 64: 219-232, 2010] conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as dense graphs, regular graphs, and trees including caterpillars and $k$-ary trees. In this note, we prove that every lobster admits an antimagic orientation.

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arXiv2020

Proof of the Core Conjecture of Hilton and Zhao

Let $G$ be a simple graph with maximum degree $Δ$. We call $G$ \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. It is NP-complete to determine the chromatic index for a general graph. However, if $G$ is overfull then $χ'(G)=Δ+1$. Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ if and only if $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. This conjecture, if true, implies an easy approach for calculating $χ'(G)$ for graphs $G$ satisfying the conditions. The progress on the conjecture has been slow: it was only confirmed for $Δ=3,4$, respectively, in 2003 and 2017. In this paper, we confirm this conjecture for all $Δ\ge 4$.

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arXiv2020

Towards the Small Quasi-Kernel Conjecture

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, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally demicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs).

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