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 $4$.
Maximum number of spanning trees in bipartite graphs with a given diameter
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles（航空飞行器数学建模与高性能计算工信部重点实验室）；University of Niš（尼什大学）；University of Primorska（沿海大学）
For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge (1-o_n(1)) r \cdot z_k $$ where $z_k > 0$ approaches $1$ extremely fast (e.g. $z_{10}=0.999971$). The minimum degree requirement is essentially tight as for every $k \ge 2$ there are connected $n$-vertex $r$-regular graphs $G$ with $r=\lfloor n/(k+1) \rfloor -2$ for which $c_k(G)=0$. Regularity may be relaxed, replacing $r$ with the geometric mean of the de…
The spanning tree packing number of a graph $G$, denoted by $τ(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $τ(G)$ is one of the classic problems in graph theory. Cioabă and Wong initiated to investigate $τ(G)$ from spectral perspectives in 2012 and since then, $τ(G)$ has been well studied using the second largest eigenvalue of the adjacency matrix in the past decade. In this paper, we further extend the results in terms of the number of edges and the spectral radius, respectively; and prove tight sufficient conditions to guarantee $τ(G)\geq k$ with…
We show that the square of every connected $S(K_{1,4})$-free graph satisfying a matching condition has a $2$-connected spanning subgraph of maximum degree at most~$3$. Furthermore, we characterise trees whose square has a $2$-connected spanning subgraph of maximum degree at most~$k$. This generalises the results on $S(K_{1,3})$-free graphs of Henry and Vogler (1985) and Harary and Schwenk (1971), respectively.
In 2001, Komlós, Sárközy and Szemerédi proved that, for each $α>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+α)n$ contains a copy of every $n$-vertex tree with maximum degree at most $cn/\log n$. We prove the corresponding result for directed graphs. That is, for each $α>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+α)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/\log n$. …
We show that for any $\varepsilon>0$ and $Δ\in\mathbb{N}$, there exists $α>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $δ(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $αn$ contains all spanning trees with maximum degree at most $Δ$. This strengthens a result of Böttcher et al.
We show that the maximum number of triples on $n$~points, if no three triples span at most five points, is $(1\pm o(1))n^2/5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices not containing a subgraph with $k$~vertices and $s$~edges. In 1973, Brown, Erdős and Sós conjectured that the limit $\lim_{n\to \infty}n^{-2}f^{(3)}(n;k,k-2)$ exists for all~$k$. They proved this for $k=4$, where the limit is $1/6$ and the extremal examples are Steiner triple systems. We prove the conjecture for $k=5$ and show that the limit is~$1/5$. The up…
For a fixed positive $ε$, we show the existence of a constant $C_ε$ with the following property: Given a $\pm1$-edge-labeling $c:E(K_n)\to \{ -1,1\}$ of the complete graph $K_n$ with $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$ of maximum degree $Δ$, one can determine in polynomial time an isomorphic copy $F'$ of $F$ in $K_n$ with $|c(E(F'))|\leq \left(\frac{3}{4}+ε\right)Δ+C_ε.$ Our approach is based on the method of conditional expectation.
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle, and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become one of the most common techniques in proving prism-hamiltonicity. Špacapan asked whether having a spanning good even cactus is equivalent to having a hamiltonian p…
Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $ϕ:E(H)\to [m]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(H)$. We give asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/\log n)$.
