Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by exactly two discrete events: Given a tree $T$ with leaf set $L$ and edge-weights $λ: E(T)\to\mathbb{N}_0$, the non-negative integer pairwise compatibility graph $\textrm{nniPCG}(T,λ,2,2)$ has vertex set $L$ and $xy$ is an edge whenever the sum of the non-negative integer weights along the unique path from $x$ to $y$ in $T$ equals $2$. A graph $G$ has a representation as $\textrm{nniPCG}(T,λ,2,2)$ if and only if its point-determining quotient $G/\!\rthin$ is a block graph, where two vertices are in relation $\rthin$ if they have the same neighborhood in $G$. If $G$ is of this type, a labeled tree $(T,λ)$ explaining $G$ can be constructed efficiently. In addition, we consider an oriented version of this class of graphs.
Given a graph $G$, the exact distance-$p$ graph $G^{[\natural p]}$ has $V(G)$ as its vertex set, and two vertices are adjacent whenever the distance between them in $G$ equals $p$. We present formulas describing the structure of exact distance-$p$ graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-$2$ graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-$p$ graphs of hypercubes $…
Let $ϕ_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $ϕ^2_H(n)$ has been well studied in literature, but for the case $r\ge 3$, the problem that determining the value of $ϕ_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $ϕ_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $ϕ_H^r(n)$ in some special cases. In this paper, we first give th…
Exact Leaf Powers on Cycles, Ladders, Crowns, and Multipartite Block Graphs
School of Mathematics and Statistics, Hanjiang Normal University（汉江师范学院数学与统计学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
The generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $χ(H)<χ(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(χ(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.
Colouring negative exact-distance graphs of signed graphs
Zhejiang Normal University（浙江师范大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息学研究所）；Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)（分析与数学社会中心）；Instituto de Ingeniería Matemática-CIMFAV（数学工程研究所-CIMFAV）；Universidad de Valparaíso（瓦尔帕莱索大学）；Computer Science Institute of Charles University（查理大学计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET（圣路易斯应用数学研究所，圣路易斯国立大学及阿根廷国家科学技术研究委员会）；Departamento de Matemática, Universidad Nacional de San Luis（圣路易斯国立大学数学系）；Université de Lorraine, LORIA（洛林大学，洛里亚实验室）
In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
Exact Computation of the Hypergraph Turan Function for Expanded Complete 2-Graphs
令l>k>2。设k-图H^k_l是由完全2-图K_l^2通过用一组新的k-2个顶点扩大每条边而得到的。Mubayi [“A hypergraph extension of Turan's theorem”，to appear in J. Combin.日(B)]渐近地计算了Turan函数ex（n，H_l^k）。这里我们确定了对所有充分大的n，ex（n，H_l^k）的精确值，解决了Mubayi的一个猜想.
The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank of an arbitrary graph G. In particular, we use the notion of determinantal rank to transform the minimum rank problem into a system of polynomial equations that can be solved by computational tools from algebraic geometry and commutative algebra. We provide computational results, explore possibilities for improvement, and discuss how the algorithm can be ext…
The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G), $ $η(G))$, where $p(G)$, $n(G)$ and $η(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix $A(G)$, respectively. Traditionally $p(G)$ (resp. $n(G)$) is called the positive (resp. negative) inertia index of $G$. In this paper, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues …
Cycles have many interesting properties and are widely studied in many disciplines. In some areas, maximising the counts of $k$-cycles are of particular interest. A natural candidate for the construction method used to maximise the number of subgraphs $H$ in a graph $G$, is the \emph{blow-up} method. Take a graph $G$ on $n$ vertices and a pattern graph $H$ on $k$ vertices, such that $n\geq k$, the blow-up method involves an iterative process of replacing vertices in $G$ with a copy of the $k$-vertex graph $H$. In this paper, we apply the blow-up method on the family of cycles. We then present …
A graph $ G $ is said to be $ (H;k) $-vertex stable if $ G $ contains a~subgraph isomorphic to $ H $ even after removing any $ k $ of its vertices alongside with their incident edges. We will denote by $ \text{stab}(H;k) $ the minimum size among sizes of all $ (H;k) $-vertex stable graphs. In this paper we consider a~case where the structure $ H $ is a~star graph $ K_{1,r} $ and the the number of vertices in $ G $ is exact, \ie equal to $ 1 + r + k $. We will show that under the above assumptions $ \text{stab}(K_{1,r};k) $ equals either $ \frac{1}{2}(k + 1)(2r + k) $, $ \frac{1}{2}\big((r + k)…
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we gi…
Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.
Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Turán graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Turán-good. (ii) The path $P_3$ is $F$-Turán-good for $F$ with $χ(F) \geq 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$…
Given integer $k$ and a $k$-graph $F$, let $t_{k-1}(n,F)$ be the minimum integer $t$ such that every $k$-graph $H$ on $n$ vertices with codegree at least $t$ contains an $F$-factor. For integers $k\geq3$ and $0\leq\ell\leq k-1$, let $\mathcal{Y}_{k,\ell}$ be a $k$-graph with two edges that shares exactly $\ell$ vertices. Han and Zhao (JCTA, 2015) asked the following question: For all $k\ge 3$, $0\le \ell\le k-1$ and sufficiently large $n$ divisible by $2k-\ell$, determine the exact value of $t_{k-1}(n,\mathcal{Y}_{k,\ell})$. In this paper, we show that $t_{k-1}(n,\mathcal{Y}_{k,\ell})=\frac{n}…
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $χ^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite pla…
An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted $χ_d^=(G)$, is the minimum $k$ such that there exists an exact $(k,d)$-coloring of $G$. In an exact $(k,d)$-coloring, which for $d=0$ corresponds to a proper coloring, each color class induces a $d$-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on $χ_d^…
Exact counting of Euler Tours for Graphs of Bounded Treewidth
本文给出了一个简单的多项式时间算法来精确计算任意树宽有界的欧拉图的欧拉圈数。已知对于一般的图，计数k的问题是#P-完全的（Brightwell and Winkler，（Brightwell and Winkler，2005））。到目前为止，没有多项式时间算法计数欧拉旅游的任何一类图是已知的，除了非常特殊的情况下，串并行图（其中有树宽2）。
Building on previous work of the author, for each finite triangle-free graph $\mathbf{G}$, we determine the equivalence relation on the copies of $\mathbf{G}$ inside the universal homogeneous triangle-free graph, $\mathcal{H}_3$, with the smallest number of equivalence classes so that each one of the classes persists in every isomorphic subcopy of $\mathcal{H}_3$. This characterizes the exact big Ramsey degrees of $\mathcal{H}_3$. It follows that the triangle-free Henson graph is a big Ramsey structure.
Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{\lceil 3.59n\rceil}}$. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that th…
The Ollivier curvature has important applications in discrete geometry and network theory, in particular as a measure of local clustering. The Ollivier curvature is defined in terms of the Wasserstein distance which, in the discrete setting, can be regarded as an optimal solution of a particular linear programme. In certain classes of graph, this linear programme may be solved \textit{a priori} giving rise to exact combinatorial expressions for the Ollivier curvature. It has been claimed by Bhattacharya and Mukherjee (2013) that an exact expression exists for the Ollivier curvature in bipartit…
Let $P$ be a set of $n$ points in strictly convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo, Dumitrescu, Hurtado, Noy, and Urrutia [2005] and then by Dujmović and Wood [2007]. Improving on their estimates, we prove the following exact formula: $$χ(D_n) = n - \left\lfloor \sqrt{2n + \tfrac{1}{4}} - \tfrac{1}{2}\right\rfloor.$$
Given an exact covering system $S = \{a_i$ (mod $d_i$) $: 1 \leq i \leq r\}$, we introduce the corresponding exact covering system digraph (ECSD) $G_S = G(d_1n+a_1, \ldots, d_rn+a_r)$. The vertices of $G_S$ are the integers and the edges are $(n, d_in+a_i)$ for each $n \in \mathbb{Z}$ and for each congruence in the covering system. We study the structure of these directed graphs, which have finitely many components, one cycle per component, as well as indegree 1 and outdegree $r$ at each vertex. We also explore the link between ECSDs that have a single component and non-standard digital repres…
Boltzmann sampling and optimal exact-size sampling for directed acyclic graphs
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072（卡昂诺曼底大学，ENSICAEN，CNRS，诺曼底大学，GREYC联合研究实验室）
Vienna University of Technology（维也纳工业大学）；Technion - Israel Institute of Technology（以色列理工学院）
We study the exact learnability of real valued graph parameters $f$ which are known to be representable as partition functions which count the number of weighted homomorphisms into a graph $H$ with vertex weights $α$ and edge weights $β$. M. Freedman, L. Lovász and A. Schrijver have given a characterization of these graph parameters in terms of the $k$-connection matrices $C(f,k)$ of $f$. Our model of learnability is based on D. Angluin's model of exact learning using membership and equivalence queries. Given such a graph parameter $f$, the learner can ask for the values of $f$ for graphs of t…
We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied $G$-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and an edge deletion $G \setminus e$ ($e$ is a non-bridge). We obtain purely combinatorial consequences of Tutte short…
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
