Excessive Factorizations of Regular Graphs

“…Bonisoli and Cariolaro [2] introduced the concept of excessive factorization, defined the parameter χ e (G), and studied excessive factorizations of regular graphs. They posed a number of open problems and conjectures.…”

Section: Introductionmentioning

confidence: 99%

“…Any 1-factorization is an excessive factorization, but the converse is obviously not true. For example, the Petersen graph has no 1-factorization, but has an excessive factorization consisting of five 1-factors (see [2]). The graphs which admit an excessive factorization are precisely those that have a 1-factor cover, that is, those that are 1-extendable.…”

Section: Introductionmentioning

confidence: 99%

“…They posed a number of open problems and conjectures. A first question is, of course, to determine χ e (G) for any graph G. It is observed in [2] that this problem is NP-hard since, if G is regular and has even order, then χ e (G) = (G) if and only if G is 1-factorizable, and to determine whether a regular graph G is 1-factorizable is NP-complete. Therefore, we can expect to be able to determine χ e (G) only for some specific classes of graphs.…”

Section: Introductionmentioning

confidence: 99%

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On minimum sets of 1‐factors covering a complete multipartite graph

Cariolaro

2008

Journal of Graph Theory

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“…Generalizing this concept, we call the excessive factorization of G a set F of 1-factors of G whose union is E(G) and such that, subject to this condition, |F | is minimum. Excessive factorizations were introduced by Bonisoli and Cariolaro [1]. The cardinality (i.e.…”

Section: Introductionmentioning

confidence: 99%

Excessive factorizations of bipartite multigraphs

Cariolaro

Rizzi

2010

Discrete Applied Mathematics

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a b s t r a c tAn excessive factorization of a multigraph G is a set F = {F 1 , F 2 , . . . , F r } of 1-factors of G whose union is E(G) and, subject to this condition, r is minimum. The integer r is called the excessive index of G and denoted by χ e (G). We set χ e (G) = ∞ if an excessive factorization does not exist. Analogously, let m be a fixed positive integer. An excessive [m]-factorization is a set M = {M 1 , M 2 , . . . , M k } of matchings of G, all of size m, whose union is E(G) and, subject to this condition, k is minimum. The integer k is denoted by χ [m] (G) and called the excessive [m]-index of G. Again, we set χ [m] (G) = ∞ if an excessive [m]-factorization does not exist. In this paper we shall prove that, for bipartite multigraphs, both the parameters χ e and χ [m] are computable in polynomial time, and we shall obtain an efficient algorithm for finding an excessive factorization and excessive [m]-factorization, respectively, of any bipartite multigraph.

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Excessive Factorizations of Regular Graphs

Cited by 27 publications

References 13 publications

The equivalence of two conjectures of Berge and Fulkerson

The equivalence of two conjectures of Berge and Fulkerson

On minimum sets of 1‐factors covering a complete multipartite graph

Excessive factorizations of bipartite multigraphs

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