Construction and Conditions for Completely Independent Spanning Trees in Hypercubes and Regular Bipartite Graphs
R. Barabde, S. A. Mane, S. A. Kandekar
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A set of \( k \) spanning trees in a graph \( G \) is called a set of \textit{completely independent spanning trees (CISTs)} if, for every pair of vertices \( x \) and \( y \), the paths connecting \( x \) and \( y \) across different trees do not share any vertices or edges, except for \( x \) and \( y \) themselves. Hasunuma conjectured that every \(2k\)-connected graph contains exactly \(k\) completely independent spanning trees (CISTs). However, Pétérfalvi disproved this conjecture.
When \( k = 2 \), the two CISTs are called a \textit{dual-CIST}. It has been shown that determining whether a graph can have \( k \) CISTs is an NP-complete problem, even when \( k = 2 \). In $2017$, Darties et al. raised the question of whether the $6-$dimensional hypercube \( Q_6 \) can have three completely independent spanning trees (CISTs). This paper provides an answer to that question.
In this paper, we first present a necessary condition for \( k \)-regular, \( k \)-connected bipartite graphs to have \( \left\lfloor \frac{k}{2} \right\rfloor \) CISTs. We also investigate that the hypercube of dimension \( n \) cannot have \( \frac{n}{2} \) CISTs, which means Hasunuma's conjecture does not hold for the hypercube \( Q_n \) when \( n \) is an even integer \(2 < n \leq 10^7 \), except when \(n = 2^r\) and \( n \in \{161038, 215326, 2568226, 3020626, 7866046, 9115426 \} \). This result also resolves a question posed by Darties et al.
The construction of multiple CISTs on the underlying graph of a network has practical applications in ensuring the fault tolerance of data transmission. In this context, we also provide a construction for three completely independent spanning trees in the hypercube \(Q_n\) for \(n \geq 7\). Our results show that Hasunuma's conjecture holds for odd integer \(n = 7\) in \(Q_n\), but does not hold for even integer \(n = 6\).
The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for $\text{sg}(K_{n,m})$ is given, as well as a formula for the crown graphs $S_n^0$. Bounds on $\text{sg}(Q_n)$ are also discussed.
The expansion of a graph is typically associated with its spectral properties - testing whether a graph is an expander is usually done using Cheeger's inequality. One can also use multiple eigenvalues in a higher-order Cheeger's inequality to test a deeper set of properties on the graph. However Cheeger's inquality, and the higher-order Cheeger's inequality, can be imprecise tools. Recently Lee, Gharan, and Trevisan constructed the Noisy Hypercube to prove the sharpness of the gap between spectral expansion and edge expansion in the higher-order Cheeger's inequality. We are concerned with the …
Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $γ_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\ge2$. We show that $γ_c(Q_n)\sim 2^n/n$. We use Hamming codes and an "expansion" method to construct leafy spanning trees in $Q_n$.
In this paper, we consider the minimal doubly resolving set problem in Hamming graphs, hypercubes and folded hypercubes. We prove that the minimal doubly resolving set problem in hypercubes is equivalent to the coin weighing problem. Then we answer an open question on the minimal doubly resolving set problem in hypercubes. We disprove a conjecture on the metric dimension problem in folded hypercubes and give some asymptotic results for the metric dimension and the minimal doubly resolving set problems in Hamming graphs and folded hypercubes by establishing connections between these problems. U…
With the aid of hypergraph transversals it is proved that $γ_t(Q_{n+1}) = 2γ(Q_n)$, where $γ_t(G)$ and $γ(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $γ_t(G \square K_2) = 2γ(G)$. Further, we show that the bipartite condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $γ_t (G \square K_2 ) = 2γ(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.
Let T1, T2,.... Tk be spanning trees in a graph G. If for any pair of vertices u and v of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges then T1, T2,.... Tk are called edge-disjoint spanning trees in G. The design of multiple edge-disjoint spanning trees has applications to the reliable communication protocols. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. For AQn (n > 2), construction of n-1 edge-disjoint spanning trees is given the result is optimal with respect…
For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.
Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets
Faculty of Mechanical Engineering, University of Ljubljana（卢布尔雅那大学机械工程学院）；Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那大学数学物理力学研究所）；Department of Mathematics, Penn State University Beaver Campus（宾夕法尼亚州立大学比弗分校数学系）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Faculty of Chemistry and Chemical Engineering, University of Maribor（马里博尔大学化学与化学工程学院）
Let T1, T2,..., Tk be spanning trees in a graph G. If for any pair of vertices {u, v} of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges and common vertices, except the vertices u and v, then T1, T2,..., Tk are called completely independent spanning trees in G. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube possesses several embeddable properties that the hypercube and its variations do not possess. For AQn (n > 5), construction of 4 completely independent spanning trees of which two trees with diameters 2n - 5 and two trees wi…
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…
A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$ that is highly symmetric in the sense that every cycle can be derived from every other cycle just by permuting the axes. We conjecture that a similar decomposition exists for every n.
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.
A spectral condition for spanning trees with restricted degrees in bipartite graphs
School of Science, Jiangsu University of Science and Technology（江苏科技大学理学院）；School of Mathematics and Information Science, Yantai University（烟台大学数学与信息科学学院）
A long-standing conjecture on spanning trees of a hypercube states that a balanced tree on $2^n$ vertices with maximum degree at most $3$ spans the hypercube of dimension $n$ \cite{havel1986}. In this paper, we settle the conjecture for a special family of binary trees. A $0$-caterpillar is a path. For $k\geq 1$, a $k$-caterpillar is a binary tree consisting of a path with $j$-caterpillars $(0\leq j\leq k-1)$ emanating from some of the vertices on the path. A $k$-caterpillar that contains a perfect matching is said to be perfectly balanced. In this paper, we show that a perfectly balanced $2$-…
We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $Θ(\sqrt{n})$. In particular this result applies to expanders, finite tori $\mathbb{Z}_m^d$ of dimension $d \geq 5$, the hypercube $\{0,1\}^m$, and small perturbations thereof.
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.
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the co…
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.
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…
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…
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
