Covering graphs with matchings of fixed size

Cariolaro, David; Fu, Hung‐Lin

doi:10.1016/j.disc.2008.09.031

Cited by 8 publications

(34 citation statements)

References 8 publications

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“…It was proved in [2] that, for every [2]-coverable graph, χ [2] (G) = max{ |E(G)|/2 , χ (G)}. To prove that we can compute χ [2] (G) in polynomial time, we construct an auxiliary graph H = (E, F ), where E is the edge set of G and e, f are adjacent vertices in H if and only if e, f are independent edges of G. Now the question ''What is the minimum number of matchings of size 2 that covers G?''…”

Section: Introductionmentioning

confidence: 91%

“…Therefore χ [2] (G) can be found in polynomial time. Furthermore, it was shown in [3] that, for every [3]-coverable graph G,…”

Section: Introductionmentioning

confidence: 98%

“…Thus the following question of the second author 1 This question will be negatively answered in a forthcoming paper of the authors [6]. 2 In this paper we shall prove that, [m] (G) is finite, also produces an excessive [m]-factorization in time which is bounded by a polynomial in |V (G)| and |E(G)|. It will also be shown that, for any bipartite multigraph G, the parameter χ e (G) can be computed in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

“…To prove that we can compute χ [2] (G) in polynomial time, we construct an auxiliary graph H = (E, F ), where E is the edge set of G and e, f are adjacent vertices in H if and only if e, f are independent edges of G. Now the question ''What is the minimum number of matchings of size 2 that covers G?'' is reduced to the question ''What is the minimum number of edges in an edge cover of H?''.…”

Section: Introductionmentioning

confidence: 99%

“…The following concept, introduced by Cariolaro and Fu in [2], is a variant of the concept of excessive factorization, where the 1-factors are replaced by matchings of fixed size. Formally, let m be a positive integer.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 91%

“…Therefore χ [2] (G) can be found in polynomial time. Furthermore, it was shown in [3] that, for every [3]-coverable graph G,…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Excessive factorizations of bipartite multigraphs

Cariolaro

Rizzi

2010

Discrete Applied Mathematics

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On the Complexity of Computing the Excessive [B]‐Index of a Graph

Cariolaro

Rizzi

2015

Journal of Graph Theory

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Let B be a positive integer and let G be a simple graph. An excessive [B]‐factorization of G is a minimum set of matchings, each of size B, whose union is E(G). The number of matchings in an excessive [B]‐factorization of G (or ∞ if an excessive [B]‐factorization does not exist) is a graph parameter called the excessive [B]‐index of G and denoted by χ[B]′(G). In this article we prove that, for any fixed value of B, the parameter χ[B]′(G) can be computed in polynomial time in the size of the graph G. This solves a problem posed by one of the authors at the 21st British Combinatorial Conference.

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On the excessive[m]-index of a tree

Mazzuoccolo

2014

Discrete Applied Mathematics

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The excessive [m]-index of a graph G, denoted by χ ′[m] (G), is the minimum number of matchings of size m needed to cover the edge-set of G. We set χ ′[m] (G) = ∞ if such a cover does not exist and we call a graphThe case m = 3 is completely solved by Cariolaro and Fu in [4]. In this paper we prove a general formula to compute the excessive [4]-index of a tree and we conjecture a possible generalization for any value of m. Furthermore, we prove that such a formula does not work for the excessive [4]-index of an arbitrary graph.

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Covering graphs with matchings of fixed size

Cited by 8 publications

References 8 publications

Excessive factorizations of bipartite multigraphs

Excessive factorizations of bipartite multigraphs

On the Complexity of Computing the Excessive [B]‐Index of a Graph

On the excessive[m]-index of a tree

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