arXiv subjects

Guantao Chen

Publications and source records attributed to Guantao Chen.

At least 37 records · Page 2Linked to original sources

arXiv2022

Linear arboricity of degenerate graphs

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, $\operatorname{la}(G) \ge \lceilΔ(G)/2\rceil$ for a graph $G$ with maximum degree $Δ(G)$. On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\operatorname{la}(G) \leq \lceil(Δ(G)+1) / 2\rceil$ for every graph $ G $. This conjecture has been verified for planar graphs and graphs whose maximum degree is at most $ 6 $, or is equal to $ 8 $ or $ 10 $. Given a positive integer $k$, a graph $G$ is $k$-degenerate if it can be reduced to a trivial graph by successive removal of vertices with degree at most $k$. We prove that for any $k$-degenerate graph $G$, $\operatorname{la}(G) = \lceilΔ(G)/2 \rceil$ provided $Δ(G) \ge 2k^2 -k$.

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arXiv2022

Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

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 the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $χ'(G)$ (resp. $χ^*(G)$). Let $Δ(G)$ be the maximum degree of $G$ and let \[Γ(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\] where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{Δ(G), \, \lceil Γ(G) \rceil \}$ is a lower bound for $χ'(G)$. As shown by Seymour, $χ^*(G)=\max\{Δ(G), \, Γ(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $χ'(G) \le \max\{Δ(G)+1, \, \lceil Γ(G) \rceil\}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $χ'(G)$, so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $χ'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $Γ(G)>Δ(G)$; third, every multigraph $G$ satisfies $χ'(G)-χ^*(G) \le 1$, so FECP has a fascinating integer rounding property.

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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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arXiv2021

A note on Goldberg's conjecture on total chromatic numbers

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 $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ''(G) = χ'(G)$ if $χ'(G) \ge Δ(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.

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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

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

Improved bounds on the Ramsey number of fans

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 $n$, a \emph{fan }$F_n$ is a graph formed by $n$ triangles that share one common vertex. We show that ${9n}/{2}-5\le r(F_n)\le {11n}/{2} + 6$ for any $n$. This improves previous best bounds $r(F_n) \le 6n$ of Lin and Li and $r(F_n) \ge 4n+2$ of Zhang, Broersma and Chen.

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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

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

Laminar Tight Cuts in Matching Covered Graphs

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, Lovász and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut.~Carvalho, Lucchesi, and Murty made a stronger conjecture: given any nontrivial tight cut $C$ in a matching covered graph $G$, there exists a nontrivial ELP-cut $D$ in $G$ which does not cross $C$.~We confirm the conjecture in this paper.

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arXiv2019

On Gupta's Co-density Conjecture

Let $G=(V,E)$ be a multigraph. The {\em cover index} $ξ(G)$ 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 $δ(G)$ be the minimum degree of $G$ and let $Φ(G)$ be the {\em co-density} of $G$, defined by \[Φ(G)=\min \Big\{\frac{2|E^+(U)|}{|U|+1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\] where $E^+(U)$ is the set of all edges of $G$ with at least one end in $U$. It is easy to see that $ξ(G) \le \min\{δ(G), \lfloor Φ(G) \rfloor\}$. In 1978 Gupta proposed the following co-density conjecture: Every multigraph $G$ satisfies $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$, which is the dual version of the Goldberg-Seymour conjecture on edge-colorings of multigraphs. In this note we prove that $ξ(G)\ge \min\{δ(G)-1, \, \lfloor Φ(G) \rfloor\}$ if $Φ(G)$ is not integral and $ξ(G)\ge \min\{δ(G)-2, \, \lfloor Φ(G) \rfloor-1\}$ otherwise. We also show that this co-density conjecture implies another conjecture concerning cover index made by Gupta in 1967.

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arXiv2019

On the Space of 2-Linkages

Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\in V$ to $r_2\in V$, we define $\partial(P) = r_2-r_1$. If $R, S\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\partial P\otimes \partial Q$ with $P$ and $Q$ disjoint oriented paths of $G$ connecting vertices in $R$ and $S$, respectively. By $L(R, S)$, we denote the submodule of $\mathbb{Z}\langle R\rangle\otimes\mathbb{Z}\langle S\rangle$ consisting all $\sum_{r\in R, s\in S} c(r,s)r\otimes s$ such that $c(r,r) = 0$ for all $r\in R\cap S$, $\sum_{r\in R} c(r, s) = 0$ for all $s\in S$, and $\sum_{s\in S} c(r, s) = 0$ for all $r\in R$. In this paper, we provide, when $G$ is sufficiently connected, characterizations when $P(G; R, S)$ is a proper subset of $L(R, S)$.

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arXiv2018

Structural properties of edge-chromatic critical multigraphs

Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k \ge Δ+1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $Δ$ denotes the maximum degree of $G$. Tashkinov tree method shows that critical graphs contain a subgraph with two important properties named closed and elementary. Recently, efforts have been made in extending graphs beyond Tashkinov trees. However, these results can only keep one of the two essential properties. In this paper, we developed techniques to extend Tashkinov trees to larger subgraphs with both properties. Applying our result, we have improved almost all known results towards Goldberg's conjecture. In particular, we showed that Goldberg's conjecture holds for graph $G$ with $|V(G)| \le 39$ and $|Δ(G)| \le 39$ and Jacobsen's equivalent conjecture holds for $m \le 39$ while the previous known bound is $23$.

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arXiv2018

Independence number of edge-chromatic critical graphs

Let $G$ be a simple graph with maximum degree $Δ(G)$ and chromatic index $χ'(G)$. A classic result of Vizing indicates that either $χ'(G )=Δ(G)$ or $χ'(G )=Δ(G)+1$. The graph $G$ is called $Δ$-critical if $G$ is connected, $χ'(G )=Δ(G)+1$ and for any $e\in E(G)$, $χ'(G-e)=Δ(G)$. Let $G$ be an $n$-vertex $Δ$-critical graph. Vizing conjectured that $α(G)$, the independence number of $G$, is at most $\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $α(G)<\frac{3n}{5}$. We show that for any given $\varepsilon\in (0,1)$, there exist positive constants $d_0(\varepsilon)$ and $D_0(\varepsilon)$ such that if $G$ is an $n$-vertex $Δ$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $α(G)<(\frac{1}{2}+\varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $Δ$-critical graph with minimum degree at least $d$ and $Δ(G)\ge (d+2)^{5d+10}$, then \[ α(G) < \left. \begin{cases} \frac{7n}{12}, & \text{if $d= 3$; } \frac{4n}{7}, & \text{if $d= 4$; } \frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n<\frac{4n}{7}, & \text{if $d\ge 19$. } \end{cases} \right. \]

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arXiv2017

Hamiltonicity of edge-chromatic critical graphs

Given a graph $G$, denote by $Δ$ and $χ^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian.

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arXiv2017

Average degrees of edge-chromatic critical graphs

Given a graph $G$, denote by $Δ$, $\bar{d}$ and $χ^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$Δ$-critical, then $\bar{d}\geq Δ-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } Δ\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } Δ=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq Δ\leq64. \end{cases} \end{displaystyle} $$ This result improves the best known bound $\frac{2}{3}(Δ+2)$ obtained by Woodall in 2007 for $Δ\geq 56$. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.

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