‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

Perfect codes and optimal anticodes in the Grassman graph Gq(n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and ``r...

The bull is a graph consisting of a triangle and two vertex-disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. A graph G is perfect if for every induced subgraph H of G, the chromatic number of H equals the size of the largest complete subgraph of H. This paper describes the structure of all bull-free perfect graphs.

If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings M1, . . . ,M6 of G with the property that every edge of G is contained in exactly two of them and Berge conjectured that its edge set can be covered by 5 perfect matchings. We define τ(G) as the least number of perfect matchings allowing to cover the edge set of a bridgeless cubic graph and we study thi...

perfect matchings. (A perfect matching or 1-factor is a set of disjoint edges covering all vertices.) This generalizes a result of Voorhoeve [11] for the case k = 3, stating that any 3-regular bipartite graph with 2n vertices has at least ( 4 3) n perfect matchings. The base in (1) is best possible for any k: let αk be the largest real number such that any k-regular bipartite graph with 2n vert...

|This paper describes an algorithm for nding all the perfect matchings in a bipartite graph. By using the binary partitioning method, our algorithm requires O(c(n+m) + n 2:5 ) computational e ort and O(nm) memory storage, (where n denotes the number of vertices, m denotes the number of edges, and c denotes the number of perfect matchings in the given bipartite graph). Keywords|bipartite graph, ...

Critical to the understanding of a graph are its chromatic number and whether or not it is perfect. Here we prove when Γ(Zn), the zero-divisor graph of Zn, is perfect and show an alternative method to [D] for determining the chromatic number in those cases. We go on to determine the chromatic number for Γ(Zp[x]/〈x 〉) where p is prime and show that an isomorphism exists between this graph and Γ(...

In the present paper, the minimal proper alternating cycle (MPAC) rotation graph R(G) of perfect matchings of a plane bipartite graph G is defined. We show that an MPAC rotation graph R(G) of G is a directed rooted tree, and thus extend such a result for generalized polyhex graphs to arbitrary plane bipartite graphs. As an immediate result, we describe a one-to-one correspondence between MPAC s...

The forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The forcing spectrum of G is the collection of forcing numbers of all perfect matchings of G. In this paper, we classify the perfect matchings of a generalized Petersen graph P (n, 2) in two types, and show that the forcing spectrum is the un...

The idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics (see [1] and the references therein). Recently, the forcing on perfect matchings has been attracting more researchers attention. A forcing set of M is a subset of M contained...