On the size and structure of graphs with a constant number of 1-factors

Dudek, Andrzej; Schmitt, John R.

doi:10.1016/j.disc.2012.01.017

Cited by 5 publications

(13 citation statements)

References 9 publications

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“…The conjecture matches the lower bound in Lemma when

p = k (2 t - 1)!!

. It also matches the value of

c_{p}

for

p \leq 6

as computed in . In Section , we verify that

C_{p}

also equals

c_{p}

for

7 \leq p \leq 10

, and we give empirical evidence that the bound holds for all p .…”

Section: A Conjectured Upper Boundsupporting

confidence: 83%

“…More generally, when

Φ (G) = p

and

| E (G) | = n^{2} / 4 + c

, the Hetyei‐extension of G yields

f (n + 2, p) \geq {(n + 2)}^{2} / 4 + c

. This observation is due to Dudek and Schmitt .…”

Section: Introductionmentioning

confidence: 85%

“…The lower bound

c_{p} \geq - (p - 1) (p - 2)

given by Dudek and Schmitt implies

N_{p} \in O (p)

. The construction in Theorem shows that

c_{p}

is nonnegative.…”

Section: Extremal Chambersmentioning

confidence: 91%

“…For even n and positive integer p , Dudek and Schmitt defined

f (n, p)

to be the maximum number of edges in an n ‐vertex graph having exactly p perfect matchings. Say that such a graph with

f (n, p)

edges is p ‐extremal .…”

Section: Introductionmentioning

confidence: 99%

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Extremal Graphs With a Given Number of Perfect Matchings

Hartke

Stolee

West

et al. 2012

Journal of Graph Theory

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Let f(n,p) denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that f(n,p)=n2/4+cp for some constant cp when n is at least some constant np. For p≤6, they also determined cp and np. For fixed p, we show that the extremal graphs for all n are determined by those with O(p) vertices. As a corollary, a computer search determines cp and np for p≤10. We also present lower bounds on f(n,p) proving that cp>0 for p≥2 (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on f(n,p). Our structural results are based on Lovász's Cathedral Theorem.

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“…The conjecture matches the lower bound in Lemma when

p = k (2 t - 1)!!

. It also matches the value of

c_{p}

for

p \leq 6

as computed in . In Section , we verify that

C_{p}

also equals

c_{p}

for

7 \leq p \leq 10

, and we give empirical evidence that the bound holds for all p .…”

Section: A Conjectured Upper Boundsupporting

confidence: 83%

“…More generally, when

Φ (G) = p

and

| E (G) | = n^{2} / 4 + c

, the Hetyei‐extension of G yields

f (n + 2, p) \geq {(n + 2)}^{2} / 4 + c

. This observation is due to Dudek and Schmitt .…”

Section: Introductionmentioning

confidence: 85%

“…The lower bound

c_{p} \geq - (p - 1) (p - 2)

given by Dudek and Schmitt implies

N_{p} \in O (p)

. The construction in Theorem shows that

c_{p}

is nonnegative.…”

Section: Extremal Chambersmentioning

confidence: 91%

“…For even n and positive integer p , Dudek and Schmitt defined

f (n, p)

to be the maximum number of edges in an n ‐vertex graph having exactly p perfect matchings. Say that such a graph with

f (n, p)

edges is p ‐extremal .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extremal Graphs With a Given Number of Perfect Matchings

Hartke

Stolee

West

et al. 2012

Journal of Graph Theory

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show abstract

“…Bal et al [1] determined the maximum number of edges of a k-uniform hypergraph with a unique perfect matching, where k ≥ 3. The maximum number of edges of a graph on n vertices with exactly p perfect matchings and the corresponding extremal graphs were studied by Dudek and Schmitt [3], and Hartke et al [4].…”

mentioning

confidence: 99%

On Graphs with a Unique Perfect Matching

Wang

Shang

Yuan

2014

Graphs and Combinatorics

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If a graph has a unique perfect matching, we call it a UPM-graph. In this paper we study UPM-graphs. It was shown by Kotzig that a connected UPM-graph has a cut edge belonging to its unique perfect matching. We strengthen this result to a further structural characterization. Using the stronger result, we present a characterization of claw-free UPM-graphs, and prove that for any fixed positive integer n, the number of edges of saturated UPM-graphs on 2n vertices form an arithmetic progression from (2n +2) log 2 (n +1) −2 2+ log 2 (n+1) +n +4 to n 2 with common difference 2. For a fixed positive integer n, we determine the number of labelled UPM-trees on 2n vertices. For a bipartite UPM-graph which has maximum number of edges, we determine the number of spanning UPM-trees of it

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Isomorph-free generation of 2-connected graphs with applications

Stolee

2011

Preprint

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Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K r -saturated graphs and find the list of uniquely K 4 -saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.

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On the size and structure of graphs with a constant number of 1-factors

Cited by 5 publications

References 9 publications

Extremal Graphs With a Given Number of Perfect Matchings

Extremal Graphs With a Given Number of Perfect Matchings

On Graphs with a Unique Perfect Matching

Isomorph-free generation of 2-connected graphs with applications

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