Alen Orbanic, Marko Petkovsek, Tomaz Pisanski, Primoz Potocnik
The number of distinct maps (pre-maps) with a single vertex and valence $d$ is computed for any value of $d$. The types of maps (pre-maps) that we consider depend on whether the underlaying graph (pre-graph) is signed or unsigned and directed or undirected.
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Counting triangles in graphs without vertex disjoint odd cycles
本文确定了在禁止包含 $\ell+1$ 个顶点不相交奇圈的图中三角形副本的最大数量及其极图，推广了已有结论。
We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.
Two product formulas for counting successive vertex orderings
本研究针对完全正则图连续顶点排序计数的两个猜想，通过推导验证了完全3-均匀超图线图与完全二部3-均匀超图线图的排序数均可表示为二项式系数差或比值的乘积形式，证实了已有猜想。
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
We present a new method to count unrooted maps on the sphere up to orientation-preserving homeomorphisms. The principle, called tree-decomposition, is to deform a map into an arborescent structure whose nodes are occupied by constrained maps. Tree-decomposition turns out to be very efficient and flexible for the enumeration of constrained families of maps. In this article, the method is applied to count unrooted 2-connected maps and, more importantly, to count unrooted 3-connected maps, which correspond to the combinatorial types of oriented convex polyhedra. Our method improves significantly …
A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable unicellular maps of fixed topology. In particular, we give exact formulas for the precubic case (all vertices of degree 1 or 3), and asymptotic formulas for the general case, when…
Among all equivelar vertex-transitive maps on a given closed surface S, the automorphism groups of maps with Schläfli types {3, 7} and {7, 3} allow the highest possible order. We describe a procedure to transform all such maps into 1- or 2-orbit maps, whose symmetry type has been previously studied. In so doing we provide a procedure to determine all vertex-transitive maps with Schläfli type {3, 7} which are neither regular or chiral. We determine all such maps on surfaces with Euler characteristic -1 \geq \c{hi} \geq -40.
A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$). This paper gives formulas enumerating such triangulations (up to equivalence) on an oriented surface of given genus. {\em Une triangulation à un sommet} d'une surface orientée compacte $S$ de genre $g$ est un graphe $T\subset S$ qui a un unique sommet et dont toutes les faces (composantes connexes de $S\setminus T$) sont des triangles (incidentes à trois ar…
A threshold graph is any graph which can be constructed from the empty graph by repeatedly adding a new vertex that is either adjacent to every vertex or to no vertices. The Eulerian number $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}$ counts the number of permutations of size $n$ with exactly $k$ ascents. Implicitly Beissinger and Peled proved that the number of labeled threshold graphs on $n\ge 2$ vertices is \[\sum_{k=1}^{n-1}(n-k)\genfrac{\langle}{\rangle}{0pt}{}{n-1}{k-1}2^k.\] Their proof used generating functions. We give a direct combinatorial proof of this result.
In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.
In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied b…
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that the problems of counting even and odd k-vertex induced subgr…
Applying the enumeration of sparse set partitions, we show that the number of set systems H such that the emptyset is not in H, the total cardinality of edges in H is n, and the vertex set of H is {1, 2, ..., m}, equals (1/log(2)+o(1))^nb_n where b_n is the n-th Bell number. The same asymptotics holds if H may be a multiset. If vertex degrees in H are restricted to be at most k, the asymptotics is (1/alpha_k+o(1))^nb_n where alpha_k is the unique root of x^k/k!+...+x^1/1!-1 in (0,1].
We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.
Counting $P_3$-convex sets in graphs
本文研究了图中$P_3$-凸集的计数问题，证明其计算复杂性，并为树和阈值图设计了线性时间算法，同时为一般图设计了高效的指数时间算法。
We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$ with respect to the simple elements (permutation braids) as generators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we use matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of $n$.
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus regarding maximal chord diagrams enumerated up to rotations.
This paper provides an alternate proof to parts of the Goulden-Slofstra formula for enumerating two vertex maps by genus, which is an extension of the famous Harer-Zagier formula that computes the Euler characteristic of the moduli space of curves. This paper also shows a further simplification to the Goulden-Slofstra formula. Portions of this alternate proof will be used in a subsequent paper, where it forms a basis for a more general result that applies for a certain class of maps with an arbitrary number of vertices.
Galvin showed that for all fixed $δ$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $δ$ that admits the most independent sets is the complete bipartite graph $K_{δ,n-δ}$. He conjectured that except perhaps for some small values of $t$, the same graph yields the maximum count of independent sets of size $t$ for each possible $t$. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples $(n,δ, t)$ with $t\geq 3$, no $n$-vertex {\em bipartite} graph with minimum degree $δ$ admits more independent sets of size $t$ than…
If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. A map is called minimal if the number of vertices is minimal. We know the bounds of number of vertex orbits of semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has proved that every semi-equivelar toroidal map has a vertex-transitive cover. In this article, we prove that if a semi-equivelar map is $k$ orbital then it has a finite index $m$-orbital minimal cover for $m \le k$. We also show the existence and classification of $n$-sheeted covers of semi-eq…
Say a digraph is k-free if it has no directed cycles of length at most k, for positive integers k. Thomasse conjectured that the number of induced 3-vertex directed paths in a simple 2-free digraph on n vertices is at most (n-1)n(n+1)/15. We present an unpublished result of Bondy proving that there are at most 2n^3/25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n^3/16 holds. We also study the problem of bounding the number of (non-induced) 4-vertex paths in 3-free digraphs. We show an upper bound of 4n^4/75 using Bondy's result for Thomasse's conje…
On quasi-polynomials counting planar tight maps
本文利用平面映射的切片分解与双射方法，给出了计数具有指定面度数的平面紧映射的拟多项式显式公式，并将Tutte切片公式推广至所有非二分映射。
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers $D$. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.
Counting Various Classes of Tournament Score Sequences
本研究依托n顶点竞赛图得分序列计数的已有方法，提出可分别对自补竞赛图、强竞赛图及二者兼具的竞赛图的不同得分序列数量进行计算的算法。
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss some possible extension to maps of higher genus.
Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle multiplying map $t\mapsto dt$
本文给出集合为 $\sigma_d$-旋转且旋转数为 $p/q$ 的充要条件，并据此恢复经典结果、枚举给定轨道数的旋转集。
The slope variety of a graph is an algebraic set whose points correspond to drawings of a graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on $n$ vertices over $\mathbb{F}_2$, and the complement-reducible graphs on $n$ vertices.
For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomassé [Almost all H-free graphs have the Erdős-Hajnal property] in the case of forbidden i…
A Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus, who proved more generally that every $r$-regular $n$-vertex oriented graph $G$ (without antiparallel edges) with $r=cn$ for some fixed $c>3/8$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this paper we address the natural que…
A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs
本文精细化Kelly引理，研究计数根顶点与删除顶点重合的有根子图，证明一般不可能，但半径大于k时可计数固定高度k的多重集，并推广至边重构。
Counting cycles in planar triangulations
该研究围绕平面三角剖分图的指定长度环计数问题，推导了不同环长度范围、不同类型平面三角剖分下环数的上下界，完善了平面图环计数领域的理论结果。
At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copi…
We present recent results on the enumeration of $q$-coloured planar maps, where each monochromatic edge carries a weight $ν$. This is equivalent to weighting each map by its Tutte polynomial, or to solving the $q$-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (non-linear) differential equation. The starting point of this result is a functional equation written by Tutte in 1971, which translates into enumerative terms a simple recursive description of planar maps. The…
Let $G$ be an infinite, vertex-transitive lattice with degree $λ$ and fix a vertex on it. Consider all cycles of length exactly $l$ from this vertex to itself on $G$. Erasing loops chronologically from these cycles, what is the fraction $F_p/λ^{\ell(p)}$ of cycles of length $l$ whose last erased loop is some chosen self-avoiding polygon $p$ of length $\ell(p)$, when $l\to\infty$ ? We use combinatorial sieves to prove an exact formula for $F_p/λ^{\ell(p)}$ that we evaluate explicitly. We further prove that for all self-avoiding polygons $p$, $F_p\in\mathbb{Q}[χ]$ with $χ$ an irrational number d…
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.
Counting $k$-cycles in $5$-connected planar triangulations
该研究证明了5-连通平面三角剖分中5圈数上界为9n-50且紧，并给出k≥6时k圈数的渐近紧上界C(k)·n^{⌊k/3⌋}。
In this paper, we first generalize a theorem for counting the number of faces of an oriented embedding of a graph that passing through a given cut-edge set [S. Stahl, Trans. Amer. Math. Soc. 259 (1980), 129--145] to all surfaces. Then we extend Stahl's bounds for the average genus of the vertex-amalgamation of graphs [S. Stahl, Discrete Math. 142 (1995), 235--245] to signed graphs.
A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar toroidal maps which are not vertex-transitive. In this article, we show that semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps. More explicitly, we prove that each semi-equivelar toroidal map has a finite vertex-tran…
Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of type $[5^3, 3]$.
Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{\lfloor n^2/4\rfloor}$ for $n \geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n \geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{\lfloor n^2/4\rfloor}$ such orientations.
We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph parameters enumerating vertex colorings satisfying conditions on the multiset of colors appearing in (closed) neighborhoods. We also generalize further to enumeration problems for what we call existence homomorphisms. Here our results are substantially less complete, though we do solve some natural problems.
We present bijections for the planar cases of two counting formulas on maps that arise from the KP hierarchy (Goulden-Jackson and Carrell-Chapuy formulas), relying on a "cut-and-slide" operation. This is the first time a bijective proof is given for quadratic map-counting formulas derived from the KP hierarchy. Up to now, only the linear one-faced case was known (Harer-Zagier recurrence and Chapuy-Féray-Fusy bijection). As far as we know, this bijection is new and not equivalent to any of the well-known bijections between planar maps and tree-like objects.
A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite graphs and trees. For complete bipartite graphs, we obtain an exact formula for their shelling numbers. And for trees, we relate their shelling numbers to linear extensions of tree posets and bound shelling numbers using vertex degrees and diameter.
We introduce a class of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focusses on the case in which Φdefines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φdescribes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that the…
Counting independent sets in expanding bipartite regular graphs
本文针对满足顶点扩张性质的一般正则二部图，利用聚合物模型和簇展开给出独立集数量的渐近展开，并证明其涵盖超立方体、中间层图及有界二部正则基图的笛卡尔积，同时提供顶点扩张的强界。
Counting cographs in locally dense graphs
针对Kohayakawa等人2010年提出的关于稠密图中固定图同态副本数的猜想，研究人员证实余图属于满足该猜想的新广泛图类。
For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $λ\vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $λ$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.
Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $π=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $π$ is connected for $1\le j\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is then $Q(G, x)=\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partition p…
Transitive cornerations in maps
本文研究地图中由顶点传递自同构群保持的角化结构，推广了对称图中的循环结构概念。
Counting coloured planar maps: differential equations
本文研究按边数和单色边数计数的 q-着色平面地图，证明其生成函数满足关于边变量的多项式微分方程，并给出显式微分系统，同时将结果推广到平面三角剖分，从而在随机平面格点上求解 q-态 Potts 模型。
Counting colored planar maps: algebraicity results
本文证明当 q 为特定形式（含 q=2,3）时，适当 q 着色平面地图及 Potts 加权地图的生成函数是代数的，并推广至三角剖分，首次求解了带两个催化变量的非线性方程。
An edge coloring of the n-vertex complete graph K_n is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that the number of Gallai colorings of K_n with at most three colors is at most 7(n+1)*2^{n choose 2}, which improves the best known upper bound of \frac{3}{2} * (n-1)! * 2^{(n-1) choose 2} in [Discrete Mathematics, 2017].
A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the rema…
Gathering Information about a Graph by Counting Walks from a Single Vertex
本文研究如何通过单个顶点的游走计数唯一确定图结构，证明了谱确定且特征多项式不可约的图所有顶点都是决定性的，并证明随机图顶点可由长度至多4的闭游走计数区分，且顶点全部游走计数由最短2n个长度决定。
A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely $k$ transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's $k$-vertex-homogeneous lattice ($k \ge 4$) then what would be the sharp bounds of the number of verte…
We study the concept of \emph{compactor}, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function $F:Σ^*\to \Bbb{N}$ and a parameterization $κ: Σ^*\to \Bbb{N}$, a compactor $({\sf P},{\sf M})$ consists of a polynomial-time computable function ${\sf P}$, called \emph{condenser}, and a computable function ${\sf M}$, called \emph{extractor}, such that $F={\sf M}\circ {\sf P}$, and the condensing ${\sf P}(x)$ of $x$ has length at most $s(κ(x))$, for any input $x\in Σ^*.$ If $s$ is a polynomial function, then the compactor is said to be of poly…
Unanticipated connections between different fragments of lambda calculus and different families of embedded graphs (a.k.a. "maps") motivate the problem of enumerating $β$-normal linear lambda terms. In this brief note, it is shown (by appeal to a theorem of Arquès and Beraud) that the sequence counting isomorphism classes of $β$-normal linear lambda terms up to free exchange of adjacent lambda abstractions coincides with the sequence counting isomorphism classes of rooted maps on oriented surfaces (A000698).
We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in part…
Bipartite Determinantal Ideals and concurrent vertex maps
本文引入并发顶点映射模型，描述 bipartite determinantal ideals 初始理想的 Stanley-Reisner 复形，并研究其顶点可分解性、壳化顺序、Hilbert 级数和 $h$-多项式等性质与应用。
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