arXiv subjects

Songling Shan

Publications and source records attributed to Songling Shan.

At least 37 records · Page 2Linked to original sources

arXiv2025

2-factors in $\frac{3}{2}$-tough maximal planar graphs

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its toughness is at least $t$. Followed from the classic 1956 result of Tutte, every more than $\frac{3}{2}$-tough planar graph on at least three vertices has a 2-factor. In 1999, Owens constructed a sequence of maximal planar graphs with toughness $\frac{3}{2}-\varepsilon$ for any $\varepsilon >0$, but the graphs do not contain any 2-factor. He then posed the question of whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor. This question was recently answered affirmatively by the third author. This naturally leads to the question: under what conditions does a $\frac{3}{2}$-tough maximal planar graph contain a 2-factor? In this paper, we provide a sufficient condition for the existence of 2-factors in $\frac{3}{2}$-tough maximal planar graphs, stated as a bound on the distance between vertices of degree 3.

math.CO↗

arXiv2025

Hamilton cycles in tough $(2P_2 \cup P_1)$-free graphs

In 1973, Chvátal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. While this conjecture is still open, work has been done to confirm it for several graph classes, including all $F$-free graphs for every 5-vertex linear forest $F$ other than $P_5$ and $2P_2\cup P_1$. In this note, we show that 11-tough $(2P_2 \cup P_1)$-free graphs on at least three vertices are Hamiltonian.

math.CO↗

arXiv2025

A strengthening of a degree sequence condition for Hamiltonicity in tough graphs

Generalizing Chvátal's classic 1972 result, Hoàng proposed in 1995 the following conjecture, which strengthens Chvátal's result in terms of toughness: Let $t\ge 1$ be a positive integer and $G$ be a $t$-tough graph on $n \ge 3$ vertices with degree sequence $d_1, d_2, \dots, d_n$ in non-increasing order. Suppose for each $i\in [1, \lfloor\frac{n-1}{2} \rfloor]$, if $d_i \le i \text{ and } d_{n-i+t} < n - i $ implies $d_j + d_{n-j+t} \ge n$ for all $j\in [i+1, \lfloor\frac{n-1}{2} \rfloor]$, then $G$ is Hamiltonian. Hoàng verified the conjecture for $t=1$. In this paper, we verfity the conjecture for all $t\ge 4$. Our proof relies on a toughness closure lemma for $t\ge 4$ that we previously established. Additionally, we show that the toughness closure lemma does not hold when $t=1$.

math.CO↗

arXiv2024

Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs

Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing) edge-$k$-coloring of $G$ is an edge-$k$-coloring such that for any two distinct vertices $u$ and $v$, the set (resp. sum) of colors taken from all the edges incident with $u$ is different from that taken from all the edges incident with $v$. The vertex-distinguishing chromatic index (resp. sum-distinguishing chromatic index), denoted $χ'_{vd}(G)$ (resp. $χ'_{sd}(G)$), is the smallest value $k$ such that $G$ has a vertex-distinguishing-edge-$k$-coloring (resp. sum-distinguishing-edge-$k$-coloring). Let $G$ be a $d$-regular graph on $n$ vertices, where $n$ is even and sufficiently large. We show that $χ'_{vd}(G) =d+2$ if $d$ is arbitrarily close to $n/2$ from above, and $χ'_{sd}(G) =d+2$ if $d\ge \frac{2n}{3}$. Our first result strengthens a result of Balister et al. in 2004 for such class of regular graphs, and our second result constitutes a significant advancement in the field of sum-distinguishing edge coloring. To achieve these results, we introduce novel edge coloring results which may be of independent interest.

math.CO↗

arXiv2024

Spanning weakly even trees of graphs

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf of $T$ that has maximum degree in $G$ belongs to the same part of the bipartition of $T$. We confirm two recent conjectures of Jackson and Yoshimoto by showing that every connected graph that is not a regular bipartite graph has a spanning weakly even tree.

math.CO↗

arXiv2024

Towards the Overfull Conjecture

Let $G$ be a simple graph with maximum degree denoted as $Δ(G)$. An overfull subgraph $H$ of $G$ is a subgraph satisfying the condition $|E(H)| > Δ(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture, stating that a graph $G$ with maximum degree $Δ(G)> \frac{1}{3}|V(G)|$ has chromatic index equal to $Δ(G)$ if and only if it does not contain any overfull subgraph. The Overfull Conjecture has many implications. For example, it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $Δ(G) > \frac{1}{3}|V(G)|$, and implies several longstanding conjectures in the area of graph edge colorings. In this paper, we make the first breakthrough towards the conjecture when not imposing a minimum degree condition on the graph: for any $0<\varepsilon \le \frac{1}{14}$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\ge n_0$ vertices with $Δ(G) \ge (1-\varepsilon)n$, then the Overfull Conjecture holds for $G$. The previous best result in this direction, due to Chetwynd and Hilton from 1989, asserts the conjecture for graphs $G$ with $Δ(G) \ge |V(G)|-3$. Our result also implies the Average Degree Conjecture of Vizing from 1968 for the same class of graphs $G$.

math.CO↗

arXiv2024

A reduction of the "cycles plus $K_4$'s" problem

Let $H$ be a 2-regular graph and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. The "cycles plus $K_4$'s" problem is to show that $G$ is 4-colourable; this is a special case of the \emph{Strong Colouring Conjecture}. In this paper we reduce the "cycles plus $K_4$'s" problem to a specific 3-colourability problem. In the 3-colourability problem, vertex-disjoint triangles are glued (in a limited way) onto a disjoint union of triangles and paths of length at most 12, and we ask for 3-colourability of the resulting graph.

math.CO↗

arXiv2024

Total coloring graphs with large maximum degree

We prove that for any graph $G$, the total chromatic number of $G$ is at most $Δ(G)+2\left\lceil \frac{|V(G)|}{Δ(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result says that if $Δ(G)\ge \frac{1}{2}|V(G)|$, then $G$ has a total coloring using at most $Δ(G)+4$ colors. When $G$ is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any $0<\varepsilon <1$, there exists $n_0\in \mathbb{N}$ such that: if $G$ is an $r$-regular graph on $n \ge n_0$ vertices with $r\ge \frac{1}{2}(1+\varepsilon) n$, then $χ_T(G) \le Δ(G)+2$. This confirms the Total Coloring Conjecture for such graphs $G$.

math.CO↗

arXiv2024

Pseudo-multifan and Lollipop

A simple graph $G$ with maximum degree $Δ$ is \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 $Δ$. 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 (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.

math.CO↗

arXiv2024

A construction of a $\frac{3}{2}$-tough plane triangulation with no 2-factor

In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is hamiltonian. This result implies that every more than $\frac{3}{2}$-tough planar graph on at least three vertices is hamiltonian and so has a 2-factor. Owens in 1999 constructed non-hamiltonian maximal planar graphs of toughness arbitrarily close to $\frac{3}{2}$ and asked whether there exists a maximal non-hamiltonian planar graph of toughness exactly $\frac{3}{2}$. In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly $\frac{3}{2}$ is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by constructing a maximal $\frac{3}{2}$-tough plane graph with no 2-factor, answering the question asked by Owens as well as by Bauer, Broersma, and Schmeichel.

math.CO↗

arXiv2024

Spanning Euler Tours in Hypergraphs

Motivated by generalizations of de Bruijn cycles to various combinatorial structures (Chung, Diaconis, and Graham), we study various Euler tours in set systems. Let $\mathcal{G}$ be a hypergraph whose corank and rank are $c\geq 3$ and $k$, respetively. The minimum $t$-degree of $\mathcal{G}$ is the fewest number of edges containing every $t$-subset of vertices. An Euler tour (family, respectively) in $\mathcal{G}$ is a (family of, respectively) closed walk(s) that (jointly, respectively) traverses each edge of $\mathcal{G}$ exactly once. An Euler tour is spanning if it traverses all the vertices of $\mathcal{G}$. We show that $\mathcal{G}$ has an Euler family if its incidence graph is $(1+\lceil k/c \rceil)$-edge-connected. Provided that the number of vertices of $\mathcal{G}$ meets a reasonable lower bound, and either $2$-degree is at least $k$ or $t$-degree is at least one for $t\geq 3$, we show that $\mathcal{G}$ has a spanning Euler tour. To exhibit the usefulness of our results, we solve a number of open problems concerning ordering blocks of a design (these have applications in other fields such as erasure-correcting codes). Answering a question of Horan and Hurlbert, we show that a Steiner quadruple system of order $n$ has a (spanning) Euler tour if and only if $n\geq 8$ and $n\equiv 2,4 \pmod 6$, and we prove a similar result for all Steiner systems, as well as all designs except for 2-designs whose index $λ$ is less than the largest block size. We nearly solve a conjecture of Dewar and Stevens on the existence of universal cycles in pairwise balanced designs. Motivated by R.L. Graham's question on the existence of Hamiltonian cycles in block-intersection graphs of Steiner triple systems, we establish the Hamiltonicity of the block-intersection graph of a large family of (not necessarily uniform) designs. All our results are constructive and of polynomial time complexity.

math.CO↗

arXiv2023

An Ore-type condition for hamiltonicity in tough graphs and the extremal examples

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, and recently the author proved a general result. The result states 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. It was conjectured in the same paper that the ``$+t$" in the bound $\frac{2n}{t+1}+t-2$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $t$-tough graphs $G$ on $n\ge 3$ vertices for which $σ_2(G) = \frac{2n}{t+1}-2$ but $G$ is non-hamiltonian.

math.CO↗

arXiv2023

Triangle-degree and triangle-distinct graphs

Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.

math.CO↗

arXiv2023

On the Multigraph Overfull Conjecture

A subgraph $H$ of a multigraph $G$ is overfull if $ |E(H) | > Δ(G) \lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph version of the conjecture as follows: Let $G$ be a multigraph with maximum multiplicity $r$ and maximum degree $Δ>\frac{1}{3} r|V(G)|$. Then $G$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. In this paper, we prove the following three results toward the Multigraph Overfull Conjecture for sufficiently large and even $n$. (1) If $G$ is $k$-regular with $k\ge r(n/2+18)$, then $G$ has a 1-factorization. This result also settles a conjecture of the first author and Tipnis from 2001 up to a constant error in the lower bound of $k$. (2) If $G$ contains an overfull subgraph and $δ(G)\ge r(n/2+18)$, then $χ'(G)=\lceil χ'_f(G) \rceil$, where $χ'_f(G)$ is the fractional chromatic index of $G$. (3) If the minimum degree of $G$ is at least $(1+\varepsilon)rn/2$ for any $0<\varepsilon<1$ and $G$ contains no overfull subgraph, then $χ'(G)=Δ(G)$. The proof is based on the decomposition of multigraphs into simple graphs and we prove a slightly weak version of a conjecture due to the first author and Tipnis from 1991 on decomposing a multigraph into constrained simple graphs. The result is of independent interests.

math.CO↗

arXiv2023

A note on Gupta's co-density conjecture

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, $ξ(G)$, is the largest number of edge covers into which the edges of $G$ can be partitioned. Clearly $ξ(G) \le δ(G)$, 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 $ ξ(G) \le |E^+(U)|/ (|U|+1)/2$. Let $ρ_c(G)$, 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 $ρ_c(G)$ provides another upper bound on $ξ(G)$. Thus $ξ(G) \le \min\{δ(G), \lfloor ρ_c(G) \rfloor \}$. For a lower bound on $ξ(G)$, in 1967, Gupta conjectured that $ξ(G) \ge \min\{δ(G)-1, \lfloor ρ_c(G) \rfloor \}$. Gupta showed that the conjecture is true when $G$ is simple, and Cao et al. verified this conjecture when $ρ_c(G)$ is not an integer. In this note, we confirm the conjecture when the maximum multiplicity of $G$ is at most two or $ \min\{δ(G)-1, \lfloor ρ_c(G) \rfloor \} \le 6$.

math.CO↗

arXiv2022

Overfullness of edge-critical graphs with small minimal core degree

Let $G$ be a simple graph. Denote by $n$, $Δ(G)$ and $χ' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > Δ(G)$, and {\it 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 \emph{core} of $G$, denoted by $G_Δ$, 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 $Δ(G)\geq \frac{2}{3}n+\frac{3k}{2}$ and $δ(G_Δ)\leq k$, then $G$ is overfull.

math.CO↗

arXiv2022

The overfull conjecture on graphs of odd order and large minimum degree

Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $Δ(G)>\frac{1}{3}|V(G)|$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and $G$ be a large graph on $n$ vertices with minimum degree at least $\frac{1}{2}(1+\varepsilon)n$. It was shown that the conjecture holds for $G$ if $n$ is even. In this paper, the same result is proved if $n$ is odd. As far as we know, this is the first result on the conjecture for graphs of odd order and with a minimum degree constraint.

math.CO↗

arXiv2022

Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

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 $3/2$ is best possible. In viewing $2K_2$, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest $R$ on 5, 6, or 7 vertices, we find the sharp toughness bound $t$ such that every $t$-tough $R$-free graph on at least three vertices has a 2-factor.

math.CO↗