11 research outputs found
Cycle systems in the complete bipartite graph minus a one-factor
AbstractLet Kn,nāI denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,nāI is a collection of m-cycles whose edges partition Kn,nāI. Necessary conditions for the existence of such an m-cycle system are that mā©¾4 is even, nā©¾3 is odd, mā©½2n, and m|n(nā1). In this paper, we show these necessary conditions are sufficient except possibly in the case that mā”0(mod4) with n<m<2n
CYCLIC HAMILTONIAN CYCLE SYSTEMS OF THE COMPLETE Graph Minus A 1-factor
In this paper, we prove that cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, Kn ā I, exist if and only if n ā” 2, 4 ( mod 8) and n ļæ½ = 2p Ī± with p prime and Ī± ā„ 1
Rotation and jump distances between graphs
A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ā E(G), uw ā E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ā E(G), wxā E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least 5) and same size, G can be j-transformed into H. For every two graphs G and H of the same order and same size, the jump distance between G and H is defined as the minimum number of edge jumps required to j-transform G into H. The rotation distance between two graphs G and H of the same order and same size is the minimum number of edge rotations needed to transform G into H. The jump and rotation distances of two graphs of the same order and same size are compared. For a set S of graphs of a fixed order at least 5 and fixed size, the jump distance graph of S has S as its vertex set and where Gā and Gā in S are adjacent if and only if . A graph G is a jump distance graph if there exists a set S of graphs of the same order and same size with . Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs
Automorphism Groups with Cyclic Commutator Subgroup and Hamilton Cycles
. It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian. Key words: graph, vertex-transitive, Hamilton cycle, commutator subgroup 1 Introduction Considerable attention has been devoted to the problem of determining whether or not a connected, vertex-transitive graph X has a Hamilton cycle [1], [8], [14]. A graph X is vertex-transitive if some group G of automorphisms of X Preprint submitted to Discrete Mathematics 5 December acts transitively on V (X). If G is abelian, then it is easy to see that X has a Hamilton cycle. Thus it is natural to try to prove the same conclusion when G is "almost abelian." Recalling ..
An Alternative Unifying Measure of Welfare Gains from Risk-Sharing
Cyclic m-cycle systems of order v are constructed for all m ā„ 3, and all v ā” 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle system of Kv ā F for all m ā„ 3, and all v ā” 2(mod 2m). the electronic journal of combinatorics 10 (2003), #R38 1
On strong digraphs with a prescribed ultracenter
summary:The (directed) distance from a vertex to a vertex in a strong digraph is the length of a shortest - (directed) path in . The eccentricity of a vertex of is the distance from to a vertex furthest from in . The radius rad is the minimum eccentricity among the vertices of and the diameter diam is the maximum eccentricity. A central vertex is a vertex with eccentricity and the subdigraph induced by the central vertices is the center . For a central vertex in a strong digraph with , the central distance of is the greatest nonnegative integer such that whenever , then is in . The maximum central distance among the central vertices of is the ultraradius urad and the subdigraph induced by the central vertices with central distance urad is the ultracenter . For a given digraph , the problem of determining a strong digraph with and is studied. This problem is also considered for digraphs that are asymmetric
Cycle decompositions IV: complete directed graphs and fixed length directed cycles
We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m; where 2ā¼mā¼n