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 where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.
We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\big(3/2+o(1)\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.
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.
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…
In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph $K_{m,n}$ with $|X|=m$ and $|Y|=n$. We prove that for any given spanning forest $F$ of $K_{m,n}$ with components $T_1,T_2,\ldots,T_k$, the number of spanning trees in $K_{m,n}$ which contain all edges in $F$ is equal to $$ \frac 1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right) \left (1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ), $$ where $m_i=|V(T_i)\cap X|$ and $n_i=|V(T_i…
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behaviour is seen to generalise the number of semiregular graphs in an elegant way. This leads us to conjecture a general formula when the number of factors is vanishing compared to the number of vertices. As a corollary, we find the average number of ways to partition the edges of a random semiregular bipartite graph into spanning semiregular subgraphs in several cases. Our proof of on…
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）
In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph components. We show that the conjecture is true for a one-side regular graph (that is a graph for which all degrees of the vertices of at least one of the components are equal). We also present a new proof of the fact that the equality holds for Ferrers graphs.
The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance dis…
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.
Let $\mathcal{F}$ be a family of fixed graphs and let $d$ be large enough. For every $d$-regular graph $G$, we study the existence of a spanning $\mathcal{F}$-free subgraph of $G$ with large minimum degree. This problem is well-understood if $\mathcal{F}$ does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs.
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…
A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(β,Δ)$-graph} if it is a bipartite graph with bandwidth at most $β|W|$ and maximum degree at most $Δ$, and it also has a proper 2-coloring $χ:W\rightarrow[2]$ such that $||χ^{-1}(1)|-|χ^{-1}(2)||\leqβ|χ^{-1}(2)|$. In this paper, we prove that for every $γ>0$ and every natural number $Δ$, there exists a constant $β>0$ such that for every balanced…
In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1,2,3,..., 2n-1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|^2 /4 for any non-empty subset R \subseteq C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appears at most once. In 2010, Suzuki generalized heterochromatic graphs to f-chromatic graphs, where any color c appears at most f(c). Moreover, he presented a necessary and sufficient condition for grap…
The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/Δ}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any t…
In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly larger than $n/2$.
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.
Generalizing well-known results of Erdős and Lovász, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$, where $λ(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that…
The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning trees of a planar bipartite graph and elements of its Jacobian. In this paper, we explore an analogue of Bernardi's results, providing a canonical simply transitive group action of the Kasteleyn cokernel of a planar bipartite graph on its set of perfect matchings, when the planar bipartite graph in question is of the form $G^+$, as defined by Kenyon, Propp and Wilson.
We develop a tool for embedding almost spanning degenerate graphs of small bandwidth. As an application, we extend the blow-up lemma to degenerate graphs of small bandwidth, the bandwidth theorem to degenerate graphs, and make progress on a conjecture of Burr and Erdős on Ramsey number of degenerate graphs.
The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ …
In this paper, we show that every $2m$-partition-connected graph $G$ has a bipartite $m$-partition-connected factor $H$ such that for each vertex $v$, $d_H(v)\le \lceil \frac{3}{4}d_G(v)\rceil$. A graph $H$ is said to be $m$-partition-connected, if it contains $m$ edge-disjoint spanning trees. As an application, we conclude that tough enough graphs with appropriate number of vertices have a bipartite $m$-partition-connected factor with maximum degree at most $3m+1$. Finally, we prove that tough enough graphs of order at least $3k$ admit a bipartite connected factor whose degrees lie in the set…
A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, Böttcher, Schacht, and Taraz verified a conjecture of Bollobás and Komlós which says that for every positive $r,Δ,γ$, there exists $β$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $Δ$ which has bandwidth at most $βn$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + γ)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend t…
Let $τ(G)$ and $τ_G(e)$ be the number of spanning trees of a connected graph $G$ and the number of spanning trees of $G$ containing edge $e$. The ratio $d_{G}(e)=τ_{G}(e)/τ(G)$ is called the spanning tree edge density of $e$, or simply density of $e$. The maximum density $\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree edge dependence of $G$, or simply dependence of $G$. Given a rational number $p/q\in (0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that $d_{G}(e)=p/q$, then we say the density $p/q$ is constructible. More specially, if there exists a…
Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $λ: E(H) \rightarrow [h]$ such that $e\in E(G_{λ(e)})$ for each $e\in E(H)$. The conditions on the minimum degree $δ(\mathcal{G})=\min_{i\in[h]}\{ δ(G_i)\}$ for finding a spanning $\mathcal{G}$-transversal isomorphic to a graph $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a spanning tree with maximum degree $o(n/\log n)$ and a power of a Hamilton cycle, etc. In th…
Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges in G(n,p) such that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.
In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the zeta function of the join of two semi-regular bipartite graphs can determine each other.
For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ has at least $k_i$ (out-)neighbours in $V_{3-i}$. We prove that it is ${\cal NP}$-complete to decide whether a digraph $D$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_1$ has an out-neighbour in $V_2$ and each…
Let $G$ be a nontrivial connected graph of order $n$, and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $κ(S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ is the connectiv…
Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a \textit{$k$-factor}. In this paper, we provide spectral conditions for a (balanced bipartite) graph with minimum degree $δ$ to be $k$-extendable, and for the existence of a $k$-factor in a balanced bipartite graph, respectively. Our results generalize some previous results on perfect matchings of graphs, and extend the results in \cite{D.F} and \cite{W.L} t…
The Bandwidth theorem of Böttcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ 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 Bollobás and Komlós. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $δ(G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $χ_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $Δ(G)$, then $χ_{D}(G)\leq Δ(G)+1$. We also obtain an upper bound for $χ_{D}(G)$ where $G$ is a graph with at most one cycle. Finally, we state a relationshi…
