Binding number conditions for matching extension

Robertshaw, Andrew M.; Woodall, Douglas R.

doi:10.1016/s0012-365x(01)00192-3

Cited by 8 publications

(5 citation statements)

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“…Many results on the binding number conditions for the existence of graph factors were acquired by Nam [12], Plummer and Saito [13], Zhou [17,18], Robertshaw and Woodall [14]. Some results on factor deleted graphs see [3,4,19], and some results on factor critical graphs see [2,5,15,16,21].…”

Section: Introductionmentioning

confidence: 99%

Remarks on path factors in graphs

Zhou

2020

RAIRO-Oper. Res.

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A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A $P_{\geq n}$-factor means a path factor with each component having at least $n$ vertices. A graph $G$ is defined as a $(P_{\geq n},m)$-factor deleted graph if $G-E'$ has a $P_{\geq n}$-factor for every $E'\subseteq E(G)$ with $|E'|=m$. A graph $G$ is defined as a $(P_{\geq n},k)$-factor critical graph if after deleting any $k$ vertices of $G$ the remaining graph of $G$ admits a $P_{\geq n}$-factor. In this paper, we demonstrate that (\romannumeral1) a graph $G$ is $(P_{\geq3},m)$-factor deleted if $\kappa(G)\geq2m+1$ and $bind(G)\geq\frac{3}{2}-\frac{1}{4m+4}$; (\romannumeral2) a graph $G$ is $(P_{\geq3},k)$-factor critical if $\kappa(G)\geq k+2$ and $bind(G)\geq\frac{5+k}{4}$.

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Section: Introductionmentioning

confidence: 99%

Remarks on path factors in graphs

Zhou

2020

RAIRO-Oper. Res.

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“…There are sufficient conditions with respect to t(G) and b(G) in terms of m, n, k to ensure the existences of kextendability, E(m, n)-extendability and other matching extensions (see Chen (1995); Liu and Yu (1998); Plummer (1988aPlummer ( , 2008). Moreover, Robertshaw and Woodall (2002) proved a remarkable result that a graph with b(G) slightly greater than 4 3 ensure k-extendability if the order of G is sufficiently large. Recently, Plummer and Saito (2017) extended this result to E(m, n)-extendability.…”

Section: Introductionmentioning

confidence: 99%

“…2 Binding Number and Matching Extendability Chen (1995) proved that a graph G of even order at least 2m+2 is m-extendable if b(G) > max{m, (7m+ 13)/12}. Robertshaw and Woodall (2002) proved a stronger result (in most cases). We state their result in a simpler but slightly weaker form below.…”

Section: Introductionmentioning

confidence: 99%

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Binding Number, Toughness and General Matching Extendability in Graphs

Lu,

2018

Preprint

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A connected graph G with at least 2m + 2n + 2 vertices which contains a perfect matching is E(m, n)-extendable, if for any two sets of disjoint independent edges M and N with |M | = m and |N | = n, there is a perfect matching F in G such that M ⊆ F and N ∩ F = ∅. Similarly, a connected graph with at least n + 2k + 2 vertices is called (n, k)-extendable if for any vertex set S of size n and any matching M of size k of G − S, G − S − V (M ) contains a perfect matching. Let ε be a small positive constant, b(G) and t(G) be the binding number and toughness of a graph G. The two main theorems of this paper are: for every graph G with sufficiently large order, 1G is E(m, n)-extendable and also (n, k)-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings.

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“…Other graph properties that are guaranteed by lower bounds on bind(G) include k-extendibility [8,15], containing a k-clique [14], and having certain types of factors [12,13].…”

Section: Introductionmentioning

confidence: 99%