本文用纯组合方法研究带生成森林的 p 价平面地图计数，将生成函数表示为双超几何方程组隐式定义的级数，证明其微分代数性，并揭示随参数 u 变化在 n^{-5/2}、n^{-3} 与 n^{-3}(\ln n)^{-2} 之间的新相变和普适类。
摘要：
我们研究配备生成森林的 p 价平面地图的计数问题，其中每个面带有权重 z，森林的每个连通分支带有权重 u。等价地，我们计数配备生成树的 p 价地图，每个面带有权重 z，每条内部活跃边在 Tutte 意义下带有权重 μ:=u+1；或者计数（对偶的）p 角剖分，其配备循环沙堆构型，每个顶点带有权重 z，变量 μ:=u+1 用于记录构型的水平。该计数问题还对应于 p 角剖分上 q 态 Potts 模型的极限 q -> 0。我们的方法是纯组合的。相应的生成函数记为 F(z,u)，它由一对通过包含双超几何级数的方程组隐式定义的级数表示。我们由该方程组推导出 F(z,u) 关于 z 是微分代数的，即满足一个关于 z 的微分方程，其系数为 z 和 u 的多项式。最近已证明这一性质对更一般的 3 价地图上的 Potts 模型成立，但所用证明远为复杂且组合性较弱。对于 u >= -1，我们研究 F(z,u) 的奇异性及其第 n 项系数的相应渐近行为。对于 u>0，我们得到平面地图的标准渐近行为，具有 n^{-5/2} 的次指数项。在 u=0 处我们观察到相变，出现 n^{-3} 项。当 u∈[-1,0) 时，我们得到极为不寻常的 n^{-3}(\ln n)^{-2} 行为。据我们所知，这是平面地图的一个新的“普适类”。
英文摘要：
We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per connected component of the forest. Equivalently, we count p-valent maps equipped with a spanning tree, with a weight z per face and a weight μ:=u+1 per internally active edge, in the sense of Tutte; or the (dual) p-angulations equipped with a recurrent sandpile configuration, with a weight z per vertex and a variable μ:=u+1 that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit q -> 0 of the q-state Potts model on p-angulations. Our approach is purely combinatorial. The associated generating function, denoted F(z,u), is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that F(z,u) is differentially algebraic in z, that is, satisfies a differential equation in z with polynomial coefficients in z and u. This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For u >= -1, we study the singularities of F(z,u) and the corresponding asymptotic behaviour of its n-th coefficient. For u>0, we find the standard asymptotic behaviour of planar maps, with a subexponential term in n^{-5/2}. At u=0 we witness a phase transition with a term n^{-3}. When u\in[-1,0), we obtain an extremely unusual behaviour in n^{-3}(\ln n)^{-2}. To our knowledge, this is a new "universality class" for planar maps.
We strengthen a result of Dross, Montassier and Pinlou (2017) that the vertex set of every triangle-free planar graph can be decomposed into a set that induces a forest and a set that induces a forest with maximum degree at most $5$, showing that $5$ can be replaced by $3$.
The classical Dodgson identity can be interpreted as a quadratic identity of spanning forest polynomials, where the spanning forests used in each polynomial are defined by how three marked vertices are divided among the component trees. We prove an analogous result with four marked vertices.
We give some insight into Tutte's definition of internally and externally active edges for spanning forests. Namely we prove, that every edge subset can be constructed from the edges of exactly one spanning forest by deleting a unique subset of the internally active edges and adding a unique subset of the externally active edges.
Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.
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 provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.
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…
We consider all spanning trees of a complete simple graph $Γ$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $τ(F)$, doesn't depend on the structure of $F$ and is completely determined by the number of vertices $q_i \, (i=1, ..., m)$ in each connected component of $F$. Specifically, $τ(F) = q_1 q_2 \cdots q_m n^{m-2}$.
A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. There are also well…
We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
It is well-known that there exists a triangle-free planar graph of $n$ verticess such that the largest induced forest has size at most $\frac{5n}{8}$. Salavatipour proved that there is a forest of size at least $\frac{5n}{9.41}$ in any triangle-free planar graph of $n$ vertices. Dross, Montassier and Pinlou improved Salavatipour's bound to $\frac{5n}{9.17}$. In this work, we further improve the bound to $\frac{5n}{9}$. Our technique is inspired by the recent ideas from Lukot'ka, Maz{á}k and Zhu.
We show that if $G$ is a $d$--regular graph on $n$ vertices, then the number of spanning forests $F(G)$ satisfies $F(G)\leq d^n$. The previous best bound due to Kahale and Schulman gave $(d+1/2+O(1/d))^n$. We also have the more precise conjecture that $$F(G)^{1/n}\leq \frac{(d-1)^{d-1}}{(d^2-2d-1)^{d/2-1}}.$$ If this conjecture is true, then the expression on the right hand side is the best possible.
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.
Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F form an increasing sequence. Hallam and Sagan showed that the generating function ISF(G,t) for increasing spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,...,n is a perfect elimination order for G. We give new, purely combinatorial proofs of these results which permit us to generalize them in sever…
For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Turán problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, \[ ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2…
We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the following three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Wa…
The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.
A sharp upper bound on the number of spanning forests of regular graphs
School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth $4$ and $5$. In particular we prove that a triangle-free planar graph of order $n$ admits an induced forest of order at least $\frac{6n+7}{11}$ , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in triangle-free planar graphs, Graphs and Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order $n$ and girth at least $5$ admits an induced forest of order at least $\frac{44n+50}{69}$.
An $({\cal F},{\cal F}_d)$-partition of a graph is a vertex-partition into two sets $F$ and $F_d$ such that the graph induced by $F$ is a forest and the one induced by $F_d$ is a forest with maximum degree at most $d$. We prove that every triangle-free planar graph admits an $({\cal F},{\cal F}_5)$-partition. Moreover we show that if for some integer $d$ there exists a triangle-free planar graph that does not admit an $({\cal F},{\cal F}_d)$-partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition.
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…
We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are triangle-free planar graphs on $n$ vertices whose largest induced linear forest has order $\lceil \frac{n}{2} \rceil + 1$.
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and…
For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{ -1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$ of $F$ in $K_n$. In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$. We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq Δ(F)+1$, where $Δ(F)$ is the maximum degree of $F$. We conjecture that this bound can be improved to $|c(E(F'))|\leq (Δ(F)-1)/2$ and verify this for $F$ being t…
We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $Δ\ge 9$ can be partitioned into $\tfrac{Δ-1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $Δ$ [Vizing, 1965] and linear arboricity at most $\lceil(Δ+1)/2\rceil$ [Wu, 1999].
We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Kir{á}ly, we conclude that every $1$-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices.
An essential spanning forest of an infinite graph $G$ is a spanning forest of $G$ in which all trees have infinitely many vertices. Let $G_n$ be an increasing sequence of finite connected subgraphs of $G$ for which $\bigcup G_n=G$. Pemantle's arguments imply that the uniform measures on spanning trees of $G_n$ converge weakly to an $\operatorname {Aut}(G)$-invariant measure $μ_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $Γ\subset \operatorname {Aut}(G)$ acts quasitransitively on $G$, then $μ_G$ is the unique $Γ$-invariant measure on essential…
We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).
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…
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 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…
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…
We compare our upper bounds on the exponential growth constant $ϕ(Λ)$ characterizing the asymptotic behavior of spanning forests on Archimedean lattices $Λ$ with recently derived upper bounds. Our upper bounds on $ϕ(Λ)$, which are very close to the respective values of $ϕ(Λ)$ that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.
We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expression are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff (Laplacian) matrix. These expressions involve the matrix of maximum out forest of the digraph. Every matrix of out forests with a fixed number of arcs and the normalized matrix of out forests are represented as polynomials in the Kirchhoff matrix; w…
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…
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega(12.52^N) non-crossing spanning trees and Omega(13.61^N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Tóth. Our analysis relies on the tools of analytic …
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a sin…
The number of rooted spanning forests of bicirculant graphs
Yunnan University of Finance and Economics（云南财经大学）；Central South University（中南大学）；Qinghai Nationalities University（青海民族大学）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）
In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case.
We present the numbers of spanning forests on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are obtained. We also derive the upper bounds of the asymptotic growth constants for both $SG_d$ and $SG_{2,b}$.
We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth $g$ and size $m$ has a feedback vertex set of size at most $\frac{4m}{3g}$, improving the trivial bound of $\frac{2m}{g}$. We also prove that every $2$-connected graph with maximum degree $3$ and order $n$ has a feedback vertex set of size at most $\frac{n+2}{3}$.
