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For a multigraph $G$, $\chi'(G)$ denotes the chromatic index of $G$, $\Delta(G)$ the maximum degree of $G$, and $\Gamma(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{ odd}\right\}$. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that $\chi'(G)=\max\{\Delta(G), \Gamma(G)\}$ or $\chi'(G)=\max\{\Delta(G) + 1, \Gamma(G)\}$. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long); and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an $O(|V|^5|E|^3)$ time algorithm for finding a $\max\{\Delta(G) + 1, \Gamma(G)\}$-edge-coloring of $G$.

The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures., Comment: 11 Pages, 5 Figures Paper identical to JGT submission

In this paper, we give a linear algorithm for obtaining the Laplacian eigenvalues of a cograph. This approach is more efficient as there is no need to directly compute the eigenvalues of Laplacian matrix related to this class of graph. As an application, we use this algorithm as a tool for obtaining a closed formula for the number of spanning trees of a cograph.

Let $G$ be a multigraph. A subset $F$ of $E(G)$ is an edge cover of $G$ if every vertex of $G$ is incident to an edge of $F$. The cover index, $\xi(G)$, is the largest number of edge covers into which the edges of $G$ can be partitioned. Clearly $\xi(G) \le \delta(G)$, the minimum degree of $G$. For $U\subseteq V(G)$, denote by $E^+(U)$ the set of edges incident to a vertex of $U$. When $|U|$ is odd, to cover all the vertices of $U$, any edge cover needs to contain at least $(|U|+1)/2$ edges from $E^+(U)$, indicating $ \xi(G) \le |E^+(U)|/ (|U|+1)/2$. Let $\rho_c(G)$, the co-density of $G$, be defined as the minimum of $|E^+(U)|/((|U|+1)/2)$ ranging over all $U\subseteq V(G)$ with $|U| $ odd and at least 3. Then $\rho_c(G)$ provides another upper bound on $\xi(G)$. Thus $\xi(G) \le \min\{\delta(G), \lfloor \rho_c(G) \rfloor \}$. For a lower bound on $\xi(G)$, in 1967, Gupta conjectured that $\xi(G) \ge \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \}$. Gupta showed that the conjecture is true when $G$ is simple, and Cao et al. verified this conjecture when $\rho_c(G)$ is not an integer. In this note, we confirm the conjecture when the maximum multiplicity of $G$ is at most two or $ \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \} \le 6$.

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 $\chi'(G) = \Delta(G)+1$. The \emph{core} of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attacking the overfull conjecture. Along this direction, we in this paper show that for any integer $k\geq 2$, if $G$ is critical with $\Delta(G)\geq \frac{2}{3}n+\frac{3k}{2}$ and $\delta(G_\Delta)\leq k$, then $G$ is overfull.

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 Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\operatorname{la}(G) \leq \lceil(\Delta(G)+1) / 2\rceil$ 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 $G$, $\operatorname{la}(G) = \lceil\Delta(G)/2 \rceil$ provided $\Delta(G) \ge 2k^2 -k$., Comment: 15 pages, 1 figure

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 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 $\{1, \dots, \Delta(G)+\mu(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge coloring of $G$. Gir\~{a}o 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 $\mu(G) \ge 2$., Comment: 23 pages,4 figures

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

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 simple connected graph with $\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ 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 $\Delta=3$ in 2003, and Cranston and Rabern proved the next case $\Delta=4$ in 2019. In this paper, we give a proof of this conjecture for all $\Delta\ge 4$., Comment: This is the second split of arXiv:2004.00734, and is the sequel to arXiv:2108.03549. arXiv admin note: substantial text overlap with arXiv:2004.00734

A simple graph $G$ with maximum degree $\Delta$ is \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 $\Delta$. 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 simple connected graph with $\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex (Core Conjecture). The goal of this paper is to develop the concepts of ``pseudo-multifan'' and ``lollipop'' and study their properties in an edge colored graph. These concepts turn out to be powerful tools in edge coloring graphs with a small core degree., Comment: This is the first split of arXiv:2004.00734

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 $|E(G)|>\Delta \lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $\chi'(G) = \Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $\Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $\Delta(G) > |V(G)|/3$. In this paper, we show that any $\Delta$-critical graph $G$ is overfull if $\Delta(G) - 7\delta(G)/4\ge(3|V(G)|-17)/4$., Comment: One portion of arXiv:2005.12909 is incorporated into this paper. arXiv admin note: text overlap with arXiv:2004.00734, arXiv:2103.05171

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 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 $\Delta(G) >n/3$. 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 $\Delta(G)\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. 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 $\Delta(G)\ge \frac{n}{2}$, no other progress on this conjecture had been made. In this paper, we confirm the conjecture for graphs $G$ with $\Delta(G) \ge 0.75n$., Comment: This submission is mainly one portion of arXiv:2005.12909 with some modifications. The second portion of arXiv:2005.12909 will be incorporated into another project

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 core of $G$, denoted by $G_{\Delta}$, 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 $\Delta(G) \ge 4$, if the maximum degree of $G_{\Delta}$ is at most two, then $G$ is overfull, which in turn gives $\Delta(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_{\Delta}$ is at most two and $\Delta(G) > n/2 +1$, then $G$ is overfull.

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.

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 every proper subgraph $H$ of $G$, and is \emph{overfull} if $|E(G)|>\Delta \lfloor (|V(G)|-1)/2 \rfloor$. Clearly, overfull graphs are Class 2. Hilton and Zhao in 1997 conjectured that if $G$ is obtained from an $n$-vertex $\Delta$-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 $\Delta\ge \frac{n}{2}(\sqrt{7}-1)\approx 0.82n$. In this paper, we improve the bound on $\Delta$ from $\frac{n}{2}(\sqrt{7}-1)$ to $0.75n$. Considering the structure of $\Delta$-critical graphs with a vertex of degree 2, we also show that for an $n$-vertex $\Delta$-critical graph with $\Delta\ge \frac{3n}{4}$, 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 $\Delta$-critical graph with $\Delta>n/3$, then $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta$. Our proof techniques are new and might shed some light on attacking both of the conjectures when $\Delta$ is large., Comment: arXiv admin note: text overlap with arXiv:2004.00734

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 $\Delta+1$. It is NP-complete to determine the chromatic index for a general graph. However, if $G$ is overfull then $\chi'(G)=\Delta+1$. Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ 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 $\chi'(G)$ for graphs $G$ satisfying the conditions. The progress on the conjecture has been slow: it was only confirmed for $\Delta=3,4$, respectively, in 2003 and 2017. In this paper, we confirm this conjecture for all $\Delta\ge 4$., Comment: 49 pages, 11 figures

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 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., Comment: This version submitted to publication to JCT-B in September, 2019

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

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 $\chi'(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let \[\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},\] where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. 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 $\chi'(G)$, 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 $\chi'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$; third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a fascinating integer rounding property.

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 $\Delta$-critical graph. Vizing conjectured that $\alpha(G)$, the independence number of $G$, is at most $\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $\alpha(G)<\frac{3n}{5}$. 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 $\Delta$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $\alpha(G)<(\frac{{1}}{2}+\varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with minimum degree at least $d$ and $\Delta(G)\ge (d+2)^{5d+10}$, then \[ \alpha(G) < \left. \begin{cases} \frac{7n}{12}, & \text{if $d= 3$; } \frac{4n}{7}, & \text{if $d= 4$; } \frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n<\frac{4n}{7}, & \text{if $d\ge 19$. } \end{cases} \right. \]

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 $L(R, S)$.

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 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 $|\Delta(G)| \le 39$ and Jacobsen's equivalent conjecture holds for $m \le 39$ while the previous known bound is $23$.

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 maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian., Comment: arXiv admin note: text overlap with arXiv:1708.01279

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 edge-$\Delta$-critical, then $\bar{d}\geq \Delta-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } \Delta\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } \Delta=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq \Delta\leq64. \end{cases} \end{displaystyle} $$ This result improves the best known bound $\frac{2}{3}(\Delta +2)$ obtained by Woodall in 2007 for $\Delta \geq 56$. 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., Comment: 23 pages

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning trail with maximum degree at most $4$. In this paper, we confirm this conjecture. We also provide examples for all $t < \frac{5}{4}$ of $t$-tough graphs that do not have a spanning trail with maximum degree at most $4$.

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 of Vizing. While deciding the exact value of $\chi'$ is a classic NP-complete problem, the computing of $\chi_f'$ is in polynomial time. In fact, it is shown that if $\chi_f' > \Delta$ then $\chi_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H$ of $G$. Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that $\chi'=\lceil\chi_f'\rceil$ if $\chi'\ge \Delta+2$, which is commonly referred as Goldberg's conjecture. In this paper, we show that if $\chi' >\Delta+\sqrt[3]{\Delta/2}$ then $\chi'=\lceil\chi_f'\rceil$. The previous best known result is for graphs with $\chi'> \Delta +\sqrt{\Delta/2}$ obtained by Scheide, and by Chen, Yu and Zang, independently. It has been shown that Goldberg's conjecture is equivalent to the following conjecture of Jakobsen: {\it For any positive integer $m$ with $m\ge 3$, every graph $G$ with $\chi'>\frac{m}{m-1}\Delta+\frac{m-3}{m-1}$ satisfies $\chi'=\lceil\chi_f'\rceil$.} Jakobsen's conjecture has been verified for $m$ up to 15 by various researchers in the last four decades. We show that it is true for $m\le 23$. Moreover, we show that Goldberg's conjecture holds for graphs $G$ with $\Delta\leq 23$ or $|V(G)|\leq 23$., Comment: 21pages

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: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous "generalization" of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning Halin subgraph. In particular, it contains a spanning Halin subgraph which is also pancyclic., Comment: 25 pages, 3 figures

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 characterization for three new classes. Furthermore, we give applications of our characterizations to some forbidden subgraph problems., Comment: 21 pages, 6 figures

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 whether a graph contains a spanning Halin graph are shown to be NP-complete. It was conjectured by Albertson, Berman, Hutchinson, and Thomassen in 1990 that a {\it every surface triangulation of at least four vertices contains a HIST}\,(confirmed). And it was conjectured by Lov\'asz and Plummer that {\it every 4-connected plane triangulation contains a spanning Halin graph}\,(disproved). Balancing the above two facts, in this paper, we consider generalized Halin graphs, a family of graph structures which are "stronger" than HISTs but "weaker" than Halin graphs in the sense of their construction constraints. To be exact, a generalized Halin graph is formed from a HIST by connecting its leaves into a cycle. Since a generalized Halin graph needs not to be planar, we investigate the minimum degree condition for a graph to contain it as a spanning subgraph. We show that there exists a positive integer $n_0$ such that any 3-connected graph with $n\ge n_0$ vertices and minimum degree at least $(2n+3)/5$ contains a spanning generalized Halin graph. As an application, the result implies that under the same condition, the graph $G$ contains a wheel-minor of order at least $n/2$. The minimum degree condition in the result is best possible.

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 hamiltonian square, in a 4-connected graph. The connectivity condition is necessary as, except $K_3$ and $K_4$, the square of a cycle is always 4-connected.

Let $G$ be a connected simple graph of order $n$ 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 $\Delta(G)+1$. Following this result, $G$ is called $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(G-e)=\Delta(G)$ for every $e\in E(G)$. In 1968, Vizing conjectured that if $G$ is an $n$-vertex $\Delta$-critical graph, then the independence number $\alpha(G)\le n/2$. Furthermore, he conjectured that, in fact, $G$ has a 2-factor. Luo and Zhao showed that if $G$ is an $n$-vertex $\Delta$-critical graph with $\Delta(G)\ge n/2$, then $\alpha(G)\le n/2$. More recently, they showed that if $G$ is an $n$-vertex $\Delta$-critical graph with $\Delta(G)\ge 6n/7$, then $G$ has a hamiltonian cycle, and so $G$ has a 2-factor. In this paper, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with $\Delta(G)\ge n/2$, then $G$ has a 2-factor.