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

Mazzuoccolo

2015

Discrete Mathematics

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MatchingChromatic index a b s t r a c t Given two positive integers l and m, with l ≤ m, an [l, m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l ≤ |M| ≤ m for everythe cardinality of M is as small as possible. The number of matchings in an excessive [l, m]-factorization of G (or ∞, if G does not admit an excessive [l, m]-factorization) is a graph parameter called the excessive [l, m]-index of G and denoted by χ ′ [l,m] (G). In this paper we study such parameter. Our main result is a general formula for the excessive [l, m]-index of a graph G in terms of other graph parameters. Furthermore, we give a polynomial time algorithm which computes χ ′ [l,m] (G) for any fixed constants l and m and outputs an excessive [l, m]-factorization of G, whenever the latter exists.

“…In Fig. 1 we show a [4,5]-covering of P consisting of 4 matchings, hence necessarily an excessive [4,5]-factorization, thereby proving that P is [4,5]-compatible and that f P (5) = 4.…”

Section: Compatibilitymentioning

confidence: 94%

“…[1,2,4,5,7,12,13,15]) and connections with some important combinatorial problems such as the Berge-Fulkerson Conjecture have already been noticed [11].…”

Section: Introductionmentioning

confidence: 92%

Excessive[l,m]-factorizations

Mazzuoccolo

2015

Discrete Mathematics

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“…Not much is known in general about the parameter χ ′ e , except that χ ′ e (G) χ ′ (G) and that the difference between χ ′ e (G) and χ ′ (G) can be arbitrarily large [1]. The present authors recently [4] determined χ ′ e (G) for all complete multipartite graphs, which proved to be a challenging task. They also introduced [3] the related notion of excessive near 1-factorization for graphs of odd order, where the size of the matchings is assumed to be the size of a near-perfect matching.…”

Section: Introductionmentioning

confidence: 91%

The Excessive [3]-Index of All Graphs

Fu²

2009

Electron. J. Combin.

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Let $m$ be a positive integer and let $G$ be a graph. A set ${\cal M}$ of matchings of $G$, all of which of size $m$, is called an $[m]$-covering of $G$ if $\bigcup_{M\in {{\cal M}}}M=E(G)$. $G$ is called $[m]$-coverable if it has an $[m]$-covering. An $[m]$-covering ${\cal M}$ such that $|{{\cal M}}|$ is minimum is called an excessive $[m]$-factorization of $G$ and the number of matchings it contains is a graph parameter called excessive $[m]$-index and denoted by $\chi'_{[m]}(G)$ (the value of $\chi'_{[m]}(G)$ is conventionally set to $\infty$ if $G$ is not $[m]$-coverable). It is obvious that $\chi'_{[1]}(G)=|E(G)|$ for every graph $G$, and it is not difficult to see that $\chi'_{[2]}(G)=\max\{\chi'(G),\lceil |E(G)|/2 \rceil\}$ for every $[2]$-coverable graph $G$. However the task of determining $\chi'_{[m]}(G)$ for arbitrary $m$ and $G$ seems to increase very rapidly in difficulty as $m$ increases, and a general formula for $m\geq 3$ is unknown. In this paper we determine such a formula for $m=3,$ thereby determining the excessive $[3]$-index for all graphs.

“…In [1] and the subsequent papers on the subject [2,4,3,5] the attention was restricted to (simple) graphs. Lemmas 1-3 were already proved in [2] in the case of graphs, and the proofs therein provided trivially extend to multigraphs.…”

Section: Some Preliminary Lemmasmentioning

confidence: 99%

Excessive factorizations of bipartite multigraphs

Discrete Applied Mathematics

Rizzi

2010

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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.