Equimatchable Regular Graphs

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3‐regular graphs are K₄ and K_{3, 3}. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r‐regular graph, then . Also it is proved that for an even r, a connected triangle‐free equimatchable r‐regular graph is isomorphic to one of the graphs C₅, C₇, and .     

Random Matchings in Regular Graphs

d -regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for let be the probability that M does not cover x. We show that for large d, the 's and the mean μ and variance of are determined to within small tolerances just by d and (in the case of μ and ) : Theorem. For any d-regular graph G, (a) , so that , (b) , where the rates of convergence depend only on d. Received: April 12, 1996     

Flows on Regular Semigroups

We study the structure of the flow monoid of a regular semigroup. This arises from the approach of Nambooripad of considering a regular semigroup as a groupoid – a category in which every morphism is invertible. A flow is then a section to the source map in this groupoid, and the monoid structure of the set of all flows is determined in terms of the Green relations on the original semigroup.     

Spreads in Strongly Regular Graphs

A spread of a strongly regular graph is a partitionof the vertex set into cliques that meet Delsarte's bound (alsocalled Hoffman's bound). Such spreads give rise to coloringsmeeting Hoffman's lower bound for the chromatic number and tocertain imprimitive three-class association schemes. These correspondenceslead to conditions for existence. Most examples come from spreadsand fans in (partial) geometries. We give other examples, includinga spread in the McLaughlin graph. For strongly regular graphsrelated to regular two-graphs, spreads give lower bounds forthe number of non-isomorphic strongly regular graphs in the switchingclass of the regular two-graph.     

Strongly Regular Semi-Cayley Graphs

We consider strongly regular graphs = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring Z G. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n = 4s ² + 4s + 2, k = 2s ² + s, = s ² – 1, and = s ²; examples are known only for s = 1, 2, and 4 (together with a noncyclic example for s = 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groups; only the first of these cases was previously known.     

Generating Random Regular Graphs

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n^1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd²). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs. * Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.     

Regular Graphs,Eigenvalues and Regular Factors

In this article, we obtain a sufficient condition for the existence of regular factors in a regular graph in terms of its third largest eigenvalue. We also determine all values of k such that every r‐regular graph with the third largest eigenvalue at most has a k‐factor.     

The Algebra of Flows in Graphs

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(K^j(X), B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges ofX; in particular, Hom(K¹(X), ) is the familiar (real) “cycle-space” ofX. We show thatK ^· (X) is torsion-free and that its Poincaré polynomial is the specializationt^{n − k}T_X(1/t, 1 + t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK ^· induces a functorial coalgebra structure onK ^· (X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K ^· (X), B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems.     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

点可迁图中的正则因子

设图G是一个K-正则连通点可迁图.如果G不是极大限制性边连通的,那么G含有一个(k-1)-因子,它的所有分支都同构于同一个阶价于k和2k-3之间的点可迁图.此结果在某种程度上加强了Watkins的相应命题:如果k正则点可迁图G不是k连通的,那么G有一个因子,它的每一个分支都同构于同一个点可迁图.     

On Circular Flows Of Graphs

A sufficient condition for graphs with circular flow index less than 4 is found in this paper. In particular, we give a simple proof of a result obtained by Galluccio and Goddyn (Combinatorica, 2002), and obtain a larger family of such graphs. * Partially supported by the National Security Agency under Grants MDA904-01-1-0022.     

Cyclotomy and Strongly Regular Graphs

We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.     

Small Directed Strongly Regular Graphs

The goal of the present paper is to provide a gallery of small directed strongly regular graphs.For each graph of order n≤12 and valency k相似文献   

Antimagic Labeling of Regular Graphs

A graph is antimagic if there is a one‐to‐one correspondence such that for any two vertices , . It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree at least three are antimagic. Whether all nonbipartite regular graphs of even degree are antimagic remained an open problem. In this article, we solve this problem and prove that all even degree regular graphs are antimagic.     

The Ferry Cover Problem on Regular Graphs and Small-Degree GraphsThe Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goatcabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

图的字典积的非零整数流(英文)

充分利用图的字典积的结构证明了以下结论:如果图G_1的每连通分支都非平凡,图G_2的阶数大于3,那么它们的字典积G_1[G_2]具有非零3-流.     

The Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.