Anthony D. Forbes, Terry S. Griggs
The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper we solve the design spectrum problem for the remaining five graphs with three possible exceptions.
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Primary Non-QE Graphs on Six Vertices
本研究针对图的二次嵌入分类问题，明确非QE图及本原非QE图的定义，完成了六个顶点所有图的QE类、非QE类、本原非QE类的完整划分。
A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed $K_3$'s, three signed $K_4$'s, and seven signed $K_5$'s. In this paper, we prove that there are sixteen signed $K_6$'s upto switching ismomorphism.
For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\m…
We prove that there are 24 4-critical $P_6$-free graphs, and give the complete list. We remark that, if $H$ is connected and not a subgraph of $P_6$, there are infinitely many 4-critical $H$-free graphs. Our result answers questions of Golovach et al. and Seymour.
Let $\cal{P}$ be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in the dual terms) that every graph $P \in \cal{P}$ is factorable into factors $P_0$, $P_1$, $P_2$ (indexed by elements of the cyclic group $Q = \{0,1,2\}$) such that every factor $P_q$ consists of two induced paths with the same length $M(q)$, and $K(q)-1$ induced cycles with the same length $2M(q)$. For $q \in Q$, we define an integer $S^+(q)$ such that the vector $(K(q), M(q), S^+(q))$ determines the graph $P$ (if $P$ is simple) uniquely up to orientation-pr…
For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing a labeling on the edges with edge-weights found by summing the labels on vertices incident to a given edge modulo $k$ so that each label appears on at most one more vertex than any other and each edge-weight appears on at most one more edge than any other. We prove that all trees are six-cordial by an adjustment of the test proposed by Hovey (1991) to show all trees are $k$-cordial.
Inspired by the work in \cite{sauer} regarding the classification of all the zero-divisor graphs with six vertices, we obtain all the zero-divisor graphs with seven vertices. Hence we classify all the zero-divisor commutative semigroups with 8 elements. We also obtain all the connected graphs with seven vertices which satisfies the necessary condition $\star$ of zero-divisor graphs given in \cite{fl} but are not the zero-divisor graphs.
Each simplicial complex and integer vector yields a vector configuration whose combinatorial properties are important for the analysis of contingency tables. We study the normality of these vector configurations including a description of operations on simplicial complexes that preserve normality, constructions of families of minimally nonnormal complexes, and computations classifying all of the normal complexes on up to six vertices. We repeat this analysis for compressed vector configurations, classifying all of the compressed complexes on up to six vertices.
We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.
Self-complementary completions on six vertices
本文证明了六顶点上自补完备化的阈值 \\(\cthreshold(6)=7\\)，并确定了八弧障碍层的五个同构类，表明普通打包弱于同阶自补完备化。
A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that strongly regular graphs with at least six vertices are $2$-reconstructible.
Let G be a plane graph and T an even subset of its vertices. It has been conjectured that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. The case k=3 is equivalent to the Four Color Theorem, and the cases k=4, which was conjectured by Seymour, and k=5 were proved by Guenin. We settle the next open case k=6.
Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we consider $gr_k(K_3 : H)$ where $H$ is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for some of them have been studied step by step in several papers. We determine all the Gallai-Ramsey numbers for the remaining graphs, and…
Small planar hypohamiltonian graphs
本文通过改造特殊子图构造出34顶点平面亚哈密顿图，并给出37、40和41顶点相关新图，改进最小阶数、三次顶点数、交叉数及自同构结果。
We present a polynomial-time algorithm that determines whether a graph that contains no induced path on six vertices and no bull (the graph with vertices a, b, c, d, e and edges ab, bc, cd, be, ce) is 4-colorable. We also show that for any fixed k the k-coloring problem can be solved in polynomial time in the class of ($P_6$, bull, gem)-free graphs.
For fixed integers p and q, let f(n,p,q) denote the minimum number of colors needed to color all of the edges of the complete graph K_n such that no clique of p vertices spans fewer than q distinct colors. A construction is given which shows that f(n,5,6) < n^(1/2+o(1)). This improves upon the best known probabilistic upper bound of O(n^(3/5)) given by Erdős and Gyárfás. It is also shown that f(n,5,6) = Ω(n^(1/2)).
The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a graph on $2p$ vertices to estimate the Euler characteristic of monotone non-evasive graph properties and get some conditions such graph properties must satisfy. We also do this by means of Oliver groups and give some lower bounds for the dimension of the simplicial complex associated to a nontrivial monotone non-evasive graph property. We apply our results to gr…
We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n tending to infinity, then the number in question is essentially the same for all values of k such that k tends to infinity but k=o(n).
Distinguishing adjacent vertices by ordering edges
本文研究通过边排序区分相邻顶点的问题，证明在特定条件下（如连通非正则图或第二类图）存在全局边序使序列区分相邻顶点，并对高度正则图给出概率性结论。
Let $Γ_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $Γ_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. The…
Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.
Cubic vertex-transitive graphs of girth six
本文完整分类了围长为6的有限简单三次点传递图，证明除Desargues图外，它们均可由环面六边形平铺、闭双曲曲面弧传递三角剖分的截断或6-正则图的弧传递二面体方案截断得到。
Graphic additive bases and graph flows
本文通过生成树打包研究 Tutte 流猜想，提出指定方向定理并证明图论加法基猜想，同时给出圆周流界及反例说明打包数大于 9/2 不足以保证模五方向。
We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible negative weights as well. For graphs with a fixed number of vertices, the result implies that there are constants $C>0$ and $n_{0}$ such that if $n\geq n_{0}$ then among all $2^{n \choose 2}$ graphs with $n$ vertices, at least $\left(1-\exp\left(-Cn\right)\right)2^{n \choose 2}$ graphs satisfy Brouwer's conjecture.
Perron-Frobenius theorem for dual tensors and its applications
本文建立对偶张量的 Perron-Frobenius 定理，利用 $M$ 矩阵广义逆给出对偶特征向量公式，并将其用于图与超图的对偶中心性，通过对偶扰动打破特征向量中心性中的顶点并列。
The main aim of this paper is to solve the design spectrum problem for Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices (including the flower snark J5), the twenty snarks on 22 vertices (including the two Loupekine snarks) and Goldberg's snark 3. Together with the Petersen graph (for which the spectrum has already been computed) this list includes all non-trivial snarks of up to 22 vertices. We also give partial results for a selection of larger graphs: the two Celmins-Swart snarks, the 26- and 34-vertex Blanusa snarks, the flower snark J7, the double star snark,…
An improved bound for 2-distance coloring of planar graphs with girth six
针对围长至少为6且最大度$Δ\neq6$的平面图，证明其2-距离色数$χ_2(G)\neqΔ+4$，改进了该类图2-距离着色的已知最优上界。
We present a polynomial-time algorithm that finds a maximum weight stable set in a graph that does not contain as an induced subgraph an induced path on six vertices or a bull (the graph with vertices $a, b, c, d, e$ and edges $ab, bc, cd, be, ce$).
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the degree polynomial sequence for some well-known graphs, we prove a theorem which gives a necessary condition for realizability of a sequence of polynomials with coefficients in positive integers. Also we calculate the degree polynomial for vertises of join, Cartesian product, tensor product, and lexicographic product of two simple graphs and for vertices of th…
We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown-Sidorenko and Exoo we determine the inducibility of K_{1,1,2} and the paw graph. The proof is obtained using semi-definite programming techniques based on a modern language of extremal graph theory, which we develop in full detail in an accessible sett…
A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in $\mathbb{R}^d$ (typically $d=2,3$) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach.
Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in R^d to admit a tensegrity with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in R^d. In particular we find surgeries on graphs that give relations between different strata. Based on numerous examples we give a description of geometric conditions defining the strata for plane tensegrities, we conjecture that the list of such conditions is sufficient to describe any stratum. We conclude the paper with p…
A short proof of the generalized Conway--Gordon--Sachs theorem
本文针对广义Conway--Gordon--Sachs定理，在整数环上给出了一个比现有证明更简短的证明，该定理将六顶点完全图的情形推广到任意n个顶点的完全图。
The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger than the maximum of 6 and the clique number of $G$. We do this by reducing the problem to imperfect ($P_6$, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactl…
In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.
Adjacency between two vertices in graphs or hypergraphs is a pairwise relationship. It is redefined in this article as 2-adjacency. In general hypergraphs, hyperedges hold for $n$-adic relationship. To keep the $n$-adic relationship the concepts of $k$-adjacency and e-adjacency are defined. In graphs 2-adjacency and e-adjacency concepts match, just as $k$-adjacency and e-adjacency do for $k$-uniform hypergraphs. For general hypergraphs these concepts are different. This paper also contributes in a uniformization process of a general hypergraph to allow the definition of an e-adjacency tensor, …
Improved bounds for the smallest 4-chromatic graph of girth six
本文将围长为6的最小4-色图的阶数界改进为29≤n₆(4)≤64，构造出64顶点的见证图并通过形式化验证，同时给出结构障碍结果。
Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\infty$ but with $l = o(n^{1/4})$. We also show, under the same conditions on $l$ and $n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.
Low order maximally single-trace graphs as the first counterexamples to large N factorization in random tensors
本文通过低阶3正则3边着色图展示了随机张量中大N极限下张量模型不变量的非因子分解性，与随机矩阵的大N因子分解形成对比。
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