75 research outputs found
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
Zero Forcing Sets and Bipartite Circulants
In this paper we introduce a class of regular bipartite graphs whose
biadjacency matrices are circulant matrices and we describe some of their
properties. Notably, we compute upper and lower bounds for the zero forcing
number for such a graph based only on the parameters that describe its
biadjacency matrix. The main results of the paper characterize the bipartite
circulant graphs that achieve equality in the lower bound.Comment: 22 pages, 13 figure
Finding Edge and Vertex Induced Cycles within Circulants.
Let H be a graph. G is a subgraph of H if V (G) ⊆ V (H) and E(G) ⊆ E(H). The subgraphs of H can be used to determine whether H is planar, a line graph, and to give information about the chromatic number. In a recent work by Beeler and Jamison [3], it was shown that it is difficult to obtain an automorphic decomposition of a triangle-free graph. As many of their examples involve circulant graphs, it is of particular interest to find triangle-free subgraphs within circulants. As a cycle with at least four vertices is a canonical example of a triangle-free subgraph, we concentrate our efforts on these. In this thesis, we will state necessary and sufficient conditions for the existence of edge induced and vertex induced cycles within circulants
Cohen-Macaulay Circulant Graphs
Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2
\rfloor}, and let I(G) denote its the edge ideal in the ring R =
k[x_1,...,x_n]. We consider the problem of determining when G is
Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay
graph G must be well-covered, we focus on known families of well-covered
circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic
circulant graphs are Cohen-Macaulay. We end with the observation that even
though the well-covered property is preserved under lexicographical products of
graphs, this is not true of the Cohen-Macaulay property.Comment: 14 page
Perfect 1-factorisations of circulants with small degree
A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed
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