We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<ε<1, there exists a constant c=c(d,ε) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-ε)n vertices with maximum degree at most d. We also prove that if an (n,D,λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λin their absolute values) has large enough spectral gap D/λas a function of d and ε, then G has a copy of every tree T as above.
Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which exte…
Nearly-uniform degree distributions in spanning subgraphs
Mathematics Institute, University of Warwick（华威大学数学研究所）；Department of Mathematics, University College London（伦敦大学学院数学系）；Department of Mathematics, ETH（苏黎世联邦理工学院数学系）
We prove that if a tree $T$ has $n$ vertices and maximum degree at most $Δ$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,Δ\log^5 n/n)$.
Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. A graph is said to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_ {1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.
We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph $G_α$ on $n$ vertices with $δ(G_α)\ge αn$ for $α>0$ and we add to it the binomial random graph $G(n,C/n)$, then with high probability the graph $G_α\cup G(n,C/n)$ contains copies of all spanning trees with maximum degree at most $Δ$ simultaneously, where $C$ depends only on $α$ and $Δ$.
Gao and Richter (1994) showed that every $3$-connected graph which embeds on the plane or the projective plane has a spanning closed walk meeting each vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995) extended this result to the torus and Klein bottle. Sanders and Zhao (2001) obtained a sharp result for higher surfaces by proving that every $3$-connected graph embeddable on a surface with Euler characteristic $χ\le -46$ admits a spanning closed walk meeting each vertex at most $\lceil \frac{6-2χ}{3}\rceil$ times. In this paper, we develop these results to the remain…
We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than $O(2{.}7156^n)$. As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by $O(147.{7}^n)$. Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by $O(6.4884^n)$, the number of plane spanning trees on a set of $n$ poin…
Let G be a simple balanced bipartite graph on $2n$ vertices, $δ= δ(G)/n$, and $ρ={δ+ \sqrt{2 δ-1} \over 2}$. If $δ> 1/2$ then it has a $\lfloor ρn \rfloor$-regular spanning subgraph. The statement is nearly tight.
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of genus $g$, improving on the known $O(n^2 + g n)$-size extended formulations following from Wong and Martin.
In this paper we construct spanning trees in hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has a bounded number of distinct rays to each boundary point. The bound depends only on the (Assouad) dimension of the boundary. As a corollary we sharpen a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree outside the graph with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We will construct a tree with these properties …
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…
Given a set of points in the Euclidean plane, the Euclidean \textit{$δ$-minimum spanning tree} ($δ$-MST) problem is the problem of finding a spanning tree with maximum degree no more than $δ$ for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean \textit{$δ$-minimum bottleneck spanning tree} ($δ$-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When $δ\leq 4$, these two problems may yield disjoint sets of optimal solutions for the same s…
The geometric $δ$-minimum spanning tree problem ($δ$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $δ$, and the sum of the lengths of the edges in the tree is minimum. The similarly defined geometric $δ$-minimum bottleneck spanning tree problem ($δ$-MBST), is the problem of finding a degree bounded spanning tree such that the length of the longest edge is minimum. For point sets that lie in the Euclidean plane, both of these problems have been shown to be NP-hard for certain specif…
We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that $P(G)$ is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erdős-Rényi random graph model $G(n,p)$. It is shown that $P(G)$ converges in probability to $e^{…
We prove several results on approximate decompositions of edge-coloured quasirandom graphs into rainbow spanning structures. More precisely, we say that an edge-colouring of a graph is locally $\ell$-bounded if no vertex is incident to more than $\ell$ edges of any given colour, and that it is (globally) $g$-bounded if no colour appears more than $g$ times in the colouring. Note that every proper colouring of an $n$-vertex graph is locally $1$-bounded, and (globally) $n/2$-bounded. Our results imply the following: (i) The existence of approximate decompositions of edge-coloured $K_n$ into rain…
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of $\mathbb{G}(n,ω(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-\varepsilon)n$ vertices, where $\varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaini…
Let $R$ and $B$ be a set of red points and a set of blue points in the plane, respectively, such that $R\cup B$ is in general position, and let $f:R \to \{2,3,4, \ldots \}$ be a function. We show that if $2\le |B|\le \sum_{x\in R}(f(x)-2) + 2$, then there exists a non-crossing geometric spanning tree $T$ on $R\cup B$ such that $2\le \operatorname{deg}_T(x)\le f(x)$ for every $x\in R$ and the set of leaves of $T$ is $B$, where every edge of $T$ is a straight-line segment.
Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.559 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given $n$ points in Euclidean space with one special point $V$, there exists a Hamiltonian path with an endpoint at $V$ that is at most 1.559 times longer than the sum of the distan…
A class of graphs closed under taking induced subgraphs is $χ$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $χ(G) \leq f(ω(G))$. We consider the following question initially studied in [A. Gy{á}rf{á}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $χ$-bounded class $\cal C$, is the class $\bar{C}$ $χ$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $χ$-bounded by the constant function $f…
In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 lo…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly $n$-edge-coloured complete bipartite graphs $K_{n,n}$ which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of $K_{n,n}$ without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constra…
We study the model $G_α\cup G(n,p)$ of randomly perturbed dense graphs, where $G_α$ is any $n$-vertex graph with minimum degree at least $αn$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every $α>0$ and $Δ\ge 5$, and every $n$-vertex graph $F$ with maximum degree at most $Δ$, we show that if $p=ω(n^{-2/(Δ+1)})$ then $G_α\cup G(n,p)$ with high probability contains …
Boettcher, Schacht and Taraz gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobas and Komlos. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result …
Let $T$ be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this paper, we give two sufficient conditions for a connected claw-free graph to have a spanning tree whose stem has a bounded number of branch vertices, and those conditions are best possible. As corollaries of main results we also give some conditions to show that a connected claw-free graph has a spanning tree whose stem is a s…
For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.
Institut für Informatik, Universität Heidelberg（海德堡大学）；Faculty of Computer Science and Mathematics, University of Passau（帕绍大学）；Department of Mathematics, London School of Economics（伦敦政治经济学院）；Zuse Institute Berlin（柏林祖斯研究所）；Department of Mathematics, University College London（伦敦大学学院）
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph …
The Bandwidth Theorem of Böttcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with Δ(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+γ)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply ou…
When $k|n$, the tree $\mathrm{Comb}_{n,k}$ consists of a path containing $n/k$ vertices, each of whose vertices has a disjoint path length $k-1$ beginning at it. We show that, for any $k=k(n)$ and $ε>0$, the binomial random graph $\mathcal{G}(n,(1+ε)\log n/ n)$ almost surely contains $\mathrm{Comb}_{n,k}$ as a subgraph. This improves a recent result of Kahn, Lubetzky and Wormald. We prove a similar statement for a more general class of trees containing both these combs and all bounded degree spanning trees which have at least $εn/ \log^9n$ disjoint bare paths length $\lceil\log^9 n\rceil$. We …
We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the strict Hadwiger number of the unit ball, i.e., the largest number of unit vectors such that the distance between any two is larger than 1.
For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $δ(H)>0$ and $Δ(H)$ denote the minimum degree and maximum degree of $H$, respectively. We prove that for all $n$ sufficiently large, if $H$ is any graph of order $n$ with $Δ(H) \le \sqrt{n}/200$, then $ex(n,H)={{n-1} \choose 2}+δ(H)-1$. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case $H=C_n$, and resolves, in a strong form, a conjecture of Glebov, Person, and Wep…
We show the existence of rainbow perfect matchings in $μn$-bounded edge colourings of Dirac bipartite graphs, for a sufficiently small $μ>0$. As an application of our results, we obtain several results on the existence of rainbow $k$-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.
Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u…
Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_k$ be a mapping with $|E(G)| \stackrel{k}{\equiv}\sum_{v\in V(G)}p(v) $. In this paper, we show that if $G$ is $(3k-3)$-edge-connected, then it has an orientation such that for each vertex $v$, $|d^+_G(v)-d_G(v)/2| < k$; also if $G$ contains $2k-2$ edge-disjoint spanning trees, then it admits such an orientation but by imposing greater out-degree bounds.
For an orientation $H$ with $n$ vertices, let $T(H)$ denote the maximum possible number of labeled copies of $H$ in an $n$-vertex tournament. It is easily seen that $T(H) \ge n!/2^{e(H)}$ as the latter is the expected number of such copies in a random tournament. For $n$ odd, let $R(H)$ denote the maximum possible number of labeled copies of $H$ in an $n$-vertex regular tournament. Adler et al. proved that, in fact, for $H=C_n$ the directed Hamilton cycle, $T(C_n) \ge (e-o(1))n!/2^{n}$ and it was observed by Alon that already $R(C_n) \ge (e-o(1))n!/2^{n}$. Similar results hold for the directed…
