For a finite undirected multigraph G=(V,E) and functions f,g:V-->\NN, let N_f^g(G,j) denote the number of (f,g)-factors of G with exactly j edges. The Heilmann-Lieb Theorem implies that \sum_j N_0^1(G,j) t^j is a polynomial with only real (negative) zeros, and hence that the sequence {N_0^1(G,j)} is strictly logarithmically concave. Separate generalizations of this theorem were obtained by Ruelle and by the author. We unify, simplify, and generalize these results by means of the Grace-Szegö-Walsh Coincidence Theorem.
We propose a universal approach to a range of enumeration problems in graphs. The key point is in contracting suitably chosen symmetric tensors placed at the vertices of a graph along the edges. In particular, this leads to an algorithm that counts the number of d-regular subgraphs of an arbitrary graph including the number of d-factors (previously we considered the case d=2 with a special emphasis on the enumeration of Hamiltonian cycles; cf. math.CO/0403339). We briefly discuss the problem of the computational complexity of this algorithm.
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these po…
This paper discusses the enumeration for the total number of all rooted spanning forests of the labeled complete tripartite graph. We enumerate the total number by a combinatorial decomposition.
We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs whose Laplacian matrices admit triangular rank-one perturbations as the class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil. Our work introduces (1) a new characterization of special 2-threshold graphs that generalizes the characterization of threshold graphs in terms of isolated and dominating vertices, and (2) a spanning tree enume…
We describe a general approach of determining the distribution of spanning subgraphs in the random graph $\G(n,p)$. In particular, we determine the distribution of spanning subgraphs of certain given degree sequences, which is a generalisation of the $d$-factors, of spanning triangle-free subgraphs, of (directed) Hamilton cycles and of spanning subgraphs that are isomorphic to a collection of vertex disjoint (directed) triangles.
We classify connected spanning convex subgraphs of the square cycles. We then show that every spanning tree of $C_n^2$ is contained in a unique nontrivial connected spanning convex subgraph of $C_n^2$. As a result, we obtain a purely combinatorial derivation of the formula for the number of spanning trees of the square cycles.
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimina…
Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)…
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.
The weighted spanning tree enumerator of a graph $G$ with weighted edges is the sum of the products of edge weights over all the spanning trees in $G$. In the special case that all of the edge weights equal $1$, the weighted spanning tree enumerator counts the number of spanning trees in $G$. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of $G$. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Mat…
In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.
Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes. By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph. We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti…
We use the subgraph replacement method to prove a simple product formula for the tilings of an 8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.
Let $D$ be a connected weighted digraph. The relation between the vertex weighted complexity (with a fixed root) of the line digraph of $D$ and the edge weighted complexity (with a fixed root) of $D$ has been given in (L. Levine, Sandpile groups and spanning trees of directed line graphs, J. Combin. Theory Ser. A 118 (2011) 350-364) and, independently, in (S. Sato, New proofs for Levine's theorems, Linear Algebra Appl. 435 (2011) 943-952). In this paper, we obtain a relation between the vertex weighted complexity of the middle digraph of $D$ and the edge weighted complexity of $D$. Particularl…
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.
Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. We say that $G$ is $l$-rigid, if it contains a spanning $l$-partition-connected subgraph $H$ with $|E(H)|=\sum_{v\in V(H)} l(v)-l(V(H))$. In this paper, we investigate decomposition of graphs into spanning partition-connected and spanning rigid subgraphs. As a consequence, we improve a recent result due…
Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $δ(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tr…
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）
Motivated by a result of [17], we determine necessary and sufficient conditions on $F\/$ with $|E(F)| \leq n-1\/$ for which $K_n - F\/$ admits a $g$-angulation. For $|E(F)| \geq n\/$, we investigate the possibility of placing $F\/$ in $K_n\/$ such that $K_n -F\/$ admits a $g$-angulation for certain families of graphs $F\/$.
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.
A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal $1$-planar. For a $3$-connected locally maximal $1$-planar graph $G$, we show the existence of a spanning $3$-connected planar subgraph and prove that $G$ is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that $G$ is traceable if $G$ has at most four $3$-vertex-cuts. Moreover, infinitely …
We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$. The second is that every graph on $n$ vertices with minimum degree $δ$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{δ+1}+2$. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of ve…
We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call "tree entropy", which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle (1993).
The domination game is played on a graph G. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy---Dominator plays so as to end the game as quickly as possible while Staller plays in such a way that the game lasts as many steps as possible. The game domination number of G is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number of G when Staller starts the game. In this paper these two games a…
Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $Δ$. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(c_{d,Δ}-O\left(\frac{1}{n}\right)\right)m(G)$, where $c_{d,Δ}=d+Ω\left(\frac{1}Δ\right)$ for fixed $d$, that is, this number visibly deviates from its expected value when considering a uniformly random copy of $G$ in $K$. For $d=…
We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-con…
Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among many other things we study $f_d=\sup_{G\in \mathcal{G}_d}F(G)^{1/v(G)}$, where $\mathcal{G}_d$ is the family of $d$--regular graphs, and $v(G)$ denotes the number of vertices of a graph $G$. We show that $f_3=2^{3/2}$, and if $(G_n)_n$ is a sequence of $3$--regular graphs with length of the shortest cycle tending to infinity, then $\lim_{n\to \infty}F(G_n)^{1…
In this paper we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs, and in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $μn$ for arbitrarily small $μ>0$. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $μ=\frac{k}{k+1}$ suffices for large $n$. For smaller values of $μ$ the given graph $G$ must satisfy additional assumptions. We show that inducing subgraphs of density $d>0$ on linear subsets of vertices and being inseparable, in the sense that every cut …
Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the Matrix Determinant Lemma and the Schur complement, can be used to elegantly count the spanning trees in several significant families of graphs.
In this paper, we consider the problem of decomposing the augmented cube $AQ_n$ into two spanning, regular, connected and pancyclic subgraphs. We prove that for $ n \geq 4$ and $ 2n - 1 = n_1 + n_2 $ with $ n_1, n_2 \geq 2,$ the augmented cube $ AQ_n$ can be decomposed into two spanning subgraphs $ H_1$ and $ H_2$ such that each $ H_i$ is $n_i$-regular and $n_i$-connected. Moreover, $H_i$ is $4$-pancyclic if $ n_i \geq 3.$
We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec diamond are always enumerated by a simple product formula. This result generalizes the previous work of Ciucu (1997) and Yang (1992) about symmetric tilings of the Aztec diamond. We also use our method to prove a closed-form product formula for the number of centrally symmetric tilings of a quasi-hexagon.
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 conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.
A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a graph-parameter $sp(G)$, that coincides with the smallest $k$ that a graph $G$ has a strongly spanning maximum $k$-edge-colorable subgraph. Our first result offers some alternative definitions of $sp(G)$. Next, we show that $Δ(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=Δ(G)$. Finally, we prove some bounds…
We calculate exponential growth constants $ϕ$ and $σ$ describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values f…
Let $G$ be a graph with $X\subseteq V(G)$ and let $l$ be an intersecting supermodular subadditive integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. Let $λ\in [0,1]$ be a real number and let $η$ be a real function on $X$. In this paper, we show that if $G$ is $l$-partition-connected and for all $S\subseteq X$, $$Θ_l(G \setminus S) \le \sum_{v\in S} (η(v) -2l(v))+l(V(G))+l(S)-λ(e_G(S))+l(S)…
