Cubic vertex-transitive graphs of girth six

In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth 6 is obtained. It is proved that every such graph, with the exception of the Desargues graph on 20 vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a 6-regular graph with respect to an arc-transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than 6 are also discussed.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

CYCLE SPACES OF GRAPHS ON THE SPHERE AND THE PROJECTIVE PLANE

Cycle base theory of a graph has been well studied in abstract mathematical field such matroid theory as Whitney and Tutte did and found many applications in pratical uses such as electric circuit theory and structure analysis, etc. In this paper graph embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph on the sphere and the projective plane and it is shown that short cycles do generate the cycle spaces in the case of ““““small face-embeddings““““. As applications the authors find the exact formulae for the minimum lengthes of cycle bases of some types of graphs and present several known results. Infinite examples shows that the conditions in their main results are best possible and there are many 3-connected planar graphs whose minimum cycle bases can not be determined by the planar formulae but may be located by re-embedding them into the projective plane.     

On the reduction of Yutsis graphs

General angular momentum recoupling coefficients can be expressed as a summation formula over products of 6-$j$ coefficients. Yutsis, Levinson and Vanagas developed graphical techniques for representing the general recoupling coefficient as a cubic graph and they describe a set of reduction rules allowing a stepwise generation of the corresponding summation formula. This paper gives an overview of the state-of-the-art heuristic algorithms, used in the latest version of our GYutsis program, for calculating general recoupling coefficients. By means of an experimental setup we show that, in particular for problems of higher order, this approach yields summation formulae which are at least as good, but are often more concise than those obtained by previous algorithms. We also give a counter-example showing that the widespread convention of reducing girth cycles first does not always lead to a shortest reduction.     

An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups

One of the remarkable contributions in the study of symmetric Cayley graphs on nonabelian simple groups is the complete classification of such graphs that are cubic and nonnormal. This naturally motivates the study of cubic (normal and nonnormal) symmetric bi-Cayley graphs on nonabelian simple groups. In this paper, the full automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for a graph being a cubic normal symmetric Cayley or bi-Cayley graph on a nonabelian simple group (one may then find many examples). As an application, we also prove that cubic symmetric Cayley graphs on nonabelian simple groups are stable.     

三度边正则图的三个无限族

图X称为边正则图，若X的自同构群Aut(X)在X的边集上的作用是正则的．本文考察了三度边正则图与四度Cayley图的关系，给出了一个由四度Cayley图构造三度边正则图的方法，并且构造了边正则图的三个无限族．     

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.     

Lock-Gain Based Graph Partitioning

We propose a new heuristic for the graph partitioning problem. Based on the traditional iterative improvement framework, the heuristic uses a new type of gain in selecting vertices to move between partitions. The new type of gain provides a good explanation for the performance difference of tie-breaking strategies in KL-based iterative improvement graph partitioning algorithms. The new heuristic performed excellently. Theoretical arguments supporting its efficacy are also provided. As the proposed heuristic is considered a good candidate for local optimization engines in metaheuristics, we combined it with a genetic algorithm as a sample case and obtained a surprising result that even the average results over 1,000 runs equalled the best known for most graphs.     

Generating unlabeled connected cubic planar graphs uniformly at random

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.     

On even cycle decompositions of line graphs of cubic graphs

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markström conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markström only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markström’s result and further prove this conjecture for cubic graphs with oddness 4.     

On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.     

Fully gated graphs: Recognition and convex operations

A graph is fully gated when every convex set of vertices is gated. Doignon posed the problem of characterizing fully gated graphs and in particular of deciding whether there is an efficient algorithm for their recognition. While the number of convex sets can be exponential, we establish that it suffices to examine only the convex hulls of pairs of vertices. This yields an elementary polynomial time algorithm for the recognition of fully gated graphs; however, it does not appear to lead to a simple structural characterization. In this direction, we establish that fully gated graphs are closed under a set of ‘convex’ operations, including a new operation which duplicates the vertices of a convex set (under some well-defined restrictions). This in turn establishes that every bipartite graph is an isometric subgraph of a fully gated graph, thereby severely limiting the potential for a characterization based on subgraphs. Finally, a large class of fully gated graphs is obtained using the presence of bipartite dominators, which suggests that simple convex operations cannot suffice to produce all fully gated graphs.     

Balanced Cayley graphs and balanced planar graphs

A balanced graph is a bipartite graph with no induced circuit of length . These graphs arise in integer linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented. The graphs in this paper are simple.     

紧图的两个结果及其应用

双随机矩阵有许多重要的应用, 紧图族可以看作是组合矩阵论中关于双随机矩阵的著名的Birkhoff定理的拓广,具有重要的研究价值． 确定一个图是否紧图是个困难的问题,目前已知的紧图族尚且不多.给出了两个重要结果：任意紧图与任意多个孤立点的不交并是紧图;任意紧图的每一个顶点上各增加一条悬挂边的图是紧图． 利用这两个结果,从已知紧图可构造出无穷多个紧图族．     

An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n²).     

Well-covered circulant graphs

A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show that several families of circulants are indeed well-covered. Since the lexicographic product of two well-covered circulants is also a well-covered circulant, our partial characterization theorems enable us to generate infinitely many families of well-covered circulants previously unknown in the literature.     

On Cubic Graphs Admitting an Edge-Transitive Solvable Group

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip₃ or the complete graph K ₄, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K _3,3, is constructed.     

Random graphons and a weak Positivstellensatz for graphs

In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2‐variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a “sum of squares” in the algebra of partially labeled graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory