Perfect 1-Factorisations of Circulant Graphs

Herke, Sarada

doi:10.14264/uql.2014.98

Cited by 2 publications

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“…In particular, we show that when k is odd

{[0, 0]}_{k - 1} > {[1, 2]}_{k - 1}

. It is straightforward to verify the result for

h = 3

and

h = 4

(for full details see ). For

h \geq 5

we show that

{[0, 0]}_{n} > {[1, 2]}_{n}

for even n by proving a stronger statement by induction.…”

Section: The Family Of Graphs Dhkmentioning

confidence: 99%

“…Case 5A: Suppose position 0 has a configuration of type

5 A (G^{2 *} B^{2 *} R Y)

as in Figure . Using standard arguments, building a coloring of positions 1 and

k - 1

from this configuration leads to a short 2‐colored cycle and hence Case 5A is not part of a P1F (for full details see ). Case 5D: Suppose position 0 is of type

5 D (B^{2} G^{2} R Y)

as in Figure .…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

“…Standard arguments considering colorings that avoid short 2‐colored cycles give rise to three possible configurations in position

k - 1

, which are all of type

5 D (R^{2} Y^{2} B G)

(see Fig. and see for the full argument). Next consider position 1.…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

“…Next consider position 1. Again, standard arguments considering colorings that avoid short 2‐colored cycles give rise to the two possible configurations in position 1 shown in Figure (for full details see ); Configuration (i) is of type

5 B (Y^{2 *} R^{2} B G)

and Configuration (ii) is of type

5 B (R^{2 *} Y^{2} B G)

. Now consider the possible combinations of configurations for positions

k - 1

and 1.…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

“…The verification of Statement 8 is routine and uses a similar technique to that of the proof of Statement 7 (for full details see [6]). Hence, for any h ≥ 3, when n is even, [0, 0] n > [1,2] n in D h,k .…”

Section: Proofmentioning

confidence: 99%

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Perfect 1-Factorizations of a Family of Cayley Graphs

Herke

Maenhaut

2014

J. Combin. Designs

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A 1‐factorization of a graph G is a decomposition of G into edge‐disjoint 1‐factors (perfect matchings), and a perfect 1‐factorization is a 1‐factorization 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‐factorizations of even order 4‐regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted Dh,k, which are Cayley graphs if and only if k is even or h=2. By solving the perfect 1‐factorization problem for a large class of Dh,k graphs, we obtain a new class of 4‐regular bipartite circulant graphs that do not have a perfect 1‐factorization, answering a problem posed in . With further study of Dh,k graphs, we prove the nonexistence of P1Fs in a class of 4‐regular non‐bipartite circulant graphs, which is further support for a conjecture made in .

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“…In particular, we show that when k is odd

{[0, 0]}_{k - 1} > {[1, 2]}_{k - 1}

. It is straightforward to verify the result for

h = 3

and

h = 4

(for full details see ). For

h \geq 5

we show that

{[0, 0]}_{n} > {[1, 2]}_{n}

for even n by proving a stronger statement by induction.…”

Section: The Family Of Graphs Dhkmentioning

confidence: 99%

“…Case 5A: Suppose position 0 has a configuration of type

5 A (G^{2 *} B^{2 *} R Y)

as in Figure . Using standard arguments, building a coloring of positions 1 and

k - 1

from this configuration leads to a short 2‐colored cycle and hence Case 5A is not part of a P1F (for full details see ). Case 5D: Suppose position 0 is of type

5 D (B^{2} G^{2} R Y)

as in Figure .…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

“…Standard arguments considering colorings that avoid short 2‐colored cycles give rise to three possible configurations in position

k - 1

, which are all of type

5 D (R^{2} Y^{2} B G)

(see Fig. and see for the full argument). Next consider position 1.…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

5 B (Y^{2 *} R^{2} B G)

and Configuration (ii) is of type

5 B (R^{2 *} Y^{2} B G)

. Now consider the possible combinations of configurations for positions

k - 1

and 1.…”

Section: Perfect 1‐factorizations Of Dhkmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Perfect 1-Factorizations of a Family of Cayley Graphs

Herke

Maenhaut

2014

J. Combin. Designs

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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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Perfect 1-Factorisations of Circulant Graphs

Cited by 2 publications

References 0 publications

Perfect 1-Factorizations of a Family of Cayley Graphs

Perfect 1-Factorizations of a Family of Cayley Graphs

Uniform cycle decompositions of complete multigraphs

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