Perfect 1-Factorizations of a Family of Cayley Graphs

Herke, Sarada; Maenhaut, Barbara

doi:10.1002/jcd.21399

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…Herke [31] constructed perfect 1-factorisations of several infinite classes of 4-regular circulant graphs, and proved that for all even n > 6 the circulant graph Circ(n, {1, 4}) does not have a perfect 1-factorisation. Herke and Maenhaut [32] defined a class of graphs, provided necessary and sufficient conditions for an element of that class to form a Cayley graph, and used that class to construct an infinite family of connected bipartite 4-regular circulant graphs of order congruent to 2 (mod 4) which do not have perfect 1-factorisations. These results do not completely characterise the set of connected 4-regular circulant graphs with perfect 1-factorisations.…”

Section: Dinitz and Dukesmentioning

confidence: 99%

Uniform cycle decompositions of complete multigraphs

Berry¹

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A decomposition of a graph G is a set {D 1 , D 2 , ..., D r } of subgraphs of G whose edge sets partition the edge set of G. A graph decomposition is uniform if the union of any two distinct subgraphs of the decomposition is isomorphic to the union of any other two distinct subgraphs, that is, D i ∪ D j ∼ = D k ∪ D l whenever 1 ≤ i < j ≤ r, 1 ≤ k < l ≤ r. This is a natural extension of the notion of uniformity of 1-factorisations (graph decompositions in which each subgraph is a 1-factor). In this project, we initiate the study of uniform decompositions of complete multigraphs into cycles, stars and paths. A complete multigraph µK n is a graph with n vertices and precisely µ edges between each pair of vertices. An m-cycle is a connected graph with m vertices in which each vertex has degree two, that is, a graph with the vertex set {v 1 , v 2 , ..., v m } and the edge set {{v 1 , v 2 }, {v 2 , v 3 }, ..., {v m−1 , v m }, {v m , v 1 }}. We show that if there exists a uniform decomposition of µK n into mcycles then (A) n = m and n ≤ 7, or (B) µ = 2 and m = n − 1, or (C) µ = 1, m = (n − 1)/2 and n ≡ 3 (mod 4) or (D) µ = 1 and 2m(m + 1) = n(n − 1). We fully characterise the complete multigraphs which admit uniform cycle decompositions in case (A). In cases (B) and (C), we construct uniform decompositions for infinitely many values of n and categorise those decompositions into isomorphism classes. In case (D) we have no examples of uniform decompositions, but we prove that the existence of such a decomposition would imply the existence of a large quasi-residual design which is not residual. We discuss the computational methods and algorithms used to examine small cases of these problems, including the proof that there is no uniform decomposition in case (A) when n = 9 or n = 15, or in case (C) when n = 15. We also present some tables of results from those algorithms, giving the uniform decompositions in case (B) when n ≤ 11. A k-star is a connected graph in which one vertex has degree k and k vertices have degree one. An m-path is a connected graph with m + 1 vertices and m edges in which two vertices have degree one and the remaining vertices have degree two, that is, a graph with vertex set {v 1 , v 2 , ..., v m , v m+1 } and edge set {{v 1 , v 2 }, {v 2 , v 3 }, ..., {v m , v m+1 }}. We show that there exists a uniform star decomposition of µK n with n > 2 precisely when µ = 2, or µ = 1 and there exists a skew Hadamard design of order n or order n−1. Finally, we prove that uniform m-path decompositions of µK n exist only when n ≤ 6, and construct all such decompositions. v Financial support

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Section: Dinitz and Dukesmentioning

confidence: 99%

Uniform cycle decompositions of complete multigraphs

Berry¹

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Multi-layer Graph Theory Utilisation for Improving Traceability and Knowledge Management in Early Design Stages

2017

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Perfect 1-factorizations

Rosa

2019

Mathematica Slovaca

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Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1. One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs. A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs. It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

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Perfect 1-Factorizations of a Family of Cayley Graphs

Cited by 3 publications

References 14 publications

Uniform cycle decompositions of complete multigraphs

Uniform cycle decompositions of complete multigraphs

Multi-layer Graph Theory Utilisation for Improving Traceability and Knowledge Management in Early Design Stages

Perfect 1-factorizations

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