15 research outputs found
Pancyclic Cayley Graphs
2010 Mathematics Subject Classification: Primary 05C25. Secondary 20K01, 05C45.Let Cay(G;S) denote the Cayley graph on a finite group G with connection set S. We extend two results about the existence of cycles in Cay(G;S) from cyclic groups to arbitrary finite Abelian groups when S is a “natural” set of generators for G.This research was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
On the distance-transitivity of the square graph of the hypercube
Let be a graph. The graph of the graph
is the graph with the vertex set in which two vertices
are adjacent if and only if their distance in is at most two. The
square graph of the hypercube has some interesting properties. For
instance, it is highly symmetric and panconnected.
In this paper, we investigate some algebraic properties of the graph
. In particular, we show that the graph is
distance-transitive. We will see that this property, in some aspects, is an
outstanding property in the class of distance-transitive graphs. We show that
the graph is an imprimitive distance-transitive graph if and only if
is an odd integer. Also, we determine the spectrum of the graph .
Moreover, we show that when is an even integer, then is an
graph, that is, is a distance-transitive primitive graph
which is not a complete or line graph.Comment: 17 pages, 1 figur