Fair 1-Factorizations and Fair Holey 1-Factorizations of Complete Multipartite Graphs

Erzurumluoğlu, Aras; Rodger, C. A.

doi:10.1007/s00373-015-1648-9

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…In what follows, G(j) denotes the subgraph of G induced by the edges colored j, d G (u) denotes the degree of vertex u in G, m G (u, v) denotes the number of edges between u and v in G, and ω(G) denotes the number of components of G. The following theorem was proved in much more generality by Bahmanian and Rodger in [1], but this is sufficient for our purposes and is stronger than the version used in [7].…”

Section: Amalgamationsmentioning

confidence: 97%

“…A constructive proof of (i), (ii) and (iii) is described in Proposition 3.3 of [7]. We use the same construction here, so again it is described explicitly.…”

Section: Proofmentioning

confidence: 99%

“…In 2002 Leach and Rodger [13] settled the existence problem for fair hamiltonian decompositions of K (n, p), showing that they exist if and only if n(p − 1) is even. In a recent paper, Erzurumluoğlu and Rodger [7] showed that fair 1-factorizations of K (n, p) exist if and only if np is even, and fair holey 1-factorizations of K (n, p) exist if and only if n(p − 1) is even and p ̸ = 2. In this paper the existence of fair holey hamiltonian decompositions of K (n, p) is completely settled (see Theorem 4.1), thereby extending results in the literature on fair graph decompositions, while simultaneously settling the existence of the largest cycle frames.…”

Section: Introductionmentioning

confidence: 99%

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Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

Erzurumluoğlu

Rodger

2015

Discrete Mathematics

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a b s t r a c tA k-factor of a graph G = (V (G), E(G)) is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K (n, p) be the complete multipartite graph with p parts, each of size n. If V 1 , . . . , V p are the p parts of V (K (n, p)), then a holey k-factor of deficiencyHence a holey k-factorization is a set of holey k-factors whose edges partition E(K (n, p)). A holey hamiltonian decomposition is a holey 2-factorization of K (n, p) where each holey 2-factor is a connected subgraph of K (n, p) − V i for some i satisfying 1 ≤ i ≤ p. A (holey) k-factorization of K (n, p) is said to be fair if the edges between each pair of parts are shared as evenly as possible among the permitted (holey) factors. In this paper the existence of fair holey hamiltonian decompositions of K (n, p) is completely settled. This result simultaneously settles the existence of cycle frames of type n p for cycles of the longest length, being a companion for results in the literature for frames with short cycle length.

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Section: Amalgamationsmentioning

confidence: 97%

“…A constructive proof of (i), (ii) and (iii) is described in Proposition 3.3 of [7]. We use the same construction here, so again it is described explicitly.…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

Erzurumluoğlu

Rodger

2015

Discrete Mathematics

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“…While the results here (Theorems 5 and 6) address general partitions, these types of questions naturally arise when edge-coloring the complete multipartite graph 1 ,..., , in which the partition is chosen to be the parts of the graph. For example, it has been shown when there exist fair equitable edge-colorings of 1 ,..., in which each color class induces a hamilton cycle [11] or a 1-factor [12].…”

Section: An Application Using Amalgamationsmentioning

confidence: 99%

On Evenly-Equitable, Balanced Edge-Colorings and Related Notions

Erzurumluoğlu

Rodger

2015

International Journal of Combinatorics

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A graph is said to be even if all vertices of have even degree. Given a -edge-coloring of a graph , for each color ∈ Z = {0, 1, . . . , − 1} let ( ) denote the spanning subgraph of in which the edge-set contains precisely the edges colored . A -edgecoloring of is said to be an -edge-coloring if for each color ∈ Z , ( ) is an even graph. A -edge-coloring of is said to be evenly-equitable if for each color ∈ Z , ( ) is an even graph, and for each vertex V ∈ ( ) and for any pair of colors , ∈ Z , |deg ( ) (V) − deg ( ) (V)| ∈ {0, 2}. For any pair of vertices {V, } let ({V, }) be the number of edges between V and in (we allow V = , where {V, V} denotes a loop incident with V). A -edge-coloring of is said to be balanced if for all pairs of colors and and all pairs of vertices V and (possibly V = ), | ( ) ({V, }) − ( ) ({V, })| ≤ 1. Hilton proved that each even graph has an evenly-equitable -edge-coloring for each ∈ N. In this paper we extend this result by finding a characterization for graphs that have an evenly-equitable, balanced -edge-coloring for each ∈ N. Correspondingly we find a characterization for even graphs to have an evenly-equitable, balanced 2-edge-coloring. Then we give an instance of how evenly-equitable, balanced edge-colorings can be used to determine if a certain fairness property of factorizations of some regular graphs is satisfied. Finally we indicate how different fairness notions on edge-colorings interact with each other.

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Amalgamations and Equitable Block-Colorings

Matson

Rodger

2018

Communications in Computer and Information Science

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Fair 1-Factorizations and Fair Holey 1-Factorizations of Complete Multipartite Graphs

Cited by 5 publications

References 11 publications

Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

On Evenly-Equitable, Balanced Edge-Colorings and Related Notions

Amalgamations and Equitable Block-Colorings

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