Highly edge‐connected regular graphs without large factorizable subgraphs

Mattiolo, Davide; Steffen, Eckhard

doi:10.1002/jgt.22729

Cited by 5 publications

(6 citation statements)

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“…For large k, we made use of Theorem 3 in [26] to find a realization with a k-factor that has many edge-disjoint 1-factors. However, this approach maybe limited since Mattiolo [18] K t I t+2 Along with requiring the connectivity of a graph G and it's complement G, Ando et al [1] showed that bounding the difference of the maximum degree and minimum degree of G would yield a 1-factor in either G or G. Ignoring the connectivity requirement we are able to show that bounding the difference d 1 − d n for a non-increasing degree sequence (d 1 . .…”

Section: Theorem 2 ([10]mentioning

confidence: 83%

On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem

Shook¹

2022

Preprint

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In 1974, Kundu showed that for even n if π = (d 1 , . . . , d n ) is a non-increasing degree sequence such that D k (π) = (d 1 − k, . . . , d n − k) is graphic, then some realization of π has a k-factor. In 1978, Brualdi and then Busch et al. in 2012, conjectured that not only is there a k-factor, but there is k-factor that can be partitioned intothen the conjecture holds. Later, Seacrest extended this to k ≤ 5. We explore this conjecture by first developing new tools that generalize edge-exchanges. With these new tools, we can drop the assumption D k (π) is graphic and show that ifthen π has a realization with k edge-disjoint 1-factors. From this we show that if d n ≥ d 1 +k−1 2 or D k (π) is graphic and d 1 ≤ max{n/2 + d n − k, (n + d n )/2}, then the conjecture holds. With a different approach we show the conjecture holds when D k (π) is graphic and d min{ n 2 ,m(π)−1} > n+3k−8 2where m(π) = max{i : d i ≥ i−1}. For r ≤ 2, Busch et al. and later Seacrest for r ≤ 4 showed that if D k (π) is graphic, then there is a realization with a k-factor whose edges can be partitioned into a (k − r)-factor and r edge-disjoint 1-factors. We improve this for any r ≤ max min{k, 4}, k+3 3. As a result, we can show that if D k (π) is graphic, then there is a realization with at least 2 k 3 edge-disjoint 1-factors.

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Section: Theorem 2 ([10]mentioning

confidence: 83%

On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem

Shook¹

2022

Preprint

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“…We also need the following lemma, whose proof is basically the same as that of Lemma 2.4 in [12] but its statement is more general. To keep the paper self-contained, the proof is presented.…”

Section: Figurementioning

confidence: 99%

“…In this section we construct a (4k +2)-edge-connected (4k +2)-graph G k without a 4k-PDPM for each integer k ≥ 1. As in [12], we first construct a graph P k by adding perfect matchings to the Petersen graph and a graph Q k by using two copies of P k . Then, we construct a graph S k and "replace" some edges of S k by copies of Q k to obtain the graph G k with the desired properties.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs

Mattiolo

Steffen

et al. 2023

SIAM J. Discrete Math.

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Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every r-edge-connected r-regular graph of even order has r − 2 pairwise disjoint perfect matchings. We show that this is not the case if r ≡ 2 mod 4. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even r.It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges.Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the 5-Cycle Double Cover Conjecture.

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“…Some of those restrictions may not be necessary as in the same paper Thomassen posed some nice conjectures and problems that lift them. However, Mattiolo [17], answering Problem 1 in [21], presented k-regular k-edge-connected graphs that cannot be partitioned into a 2-factor and k − 2 1-factors. Thus, our strategy maybe limited to finding many edge-disjoint 1-factors, but not k of them.…”

Section: Theorem 7 ([1]mentioning

confidence: 99%

Maximally Edge-Connected Realizations and Kundu's $k$-factor Theorem

Shook¹

2022

Preprint

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A simple graph G with edge-connectivity λ(G) and minimum degree δ(G) is maximally edge connected if λ(G) = δ(G). In 1964, given a non-increasing degree sequence π = (d 1 , . . . , d n ), Jack Edmonds showed that there is a realization G of π that is k-edgeconnected if and only ifthen there is a maximally edge-connected realization of π with G 0 − E(Z 0 ) as a subgraph. Our theorem tells us that there is a maximally edge-connected realization of π that differs from G 0 by at most n − 1 edges. For δ(G 0 ) ≥ 2, if G 0 has a spanning forest with c components, then our theorem says there is a maximally edge-connected realization that differs from G 0 by at most n − c edges. As an application we combine our work with Kundu's k-factor Theorem to find maximally edge-connected realizations with a (k 1 , . . . , k n )-factor for k ≤ k i ≤ k + 1 and present a partial result to a conjecture that strengthens the regular case of Kundu's k-factor theorem.

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Highly edge‐connected regular graphs without large factorizable subgraphs

Abstract: We construct highly edge‐connected r‐regular graphs of even order which do not contain r − 2 pairwise disjoint perfect matchings. When r is a multiple of 4, the result solves a problem of Thomassen [4].

Cited by 5 publications

References 5 publications

On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem

On a conjecture that strengthens the $k$-factor case of Kundu's $k$-factor Theorem

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs

Maximally Edge-Connected Realizations and Kundu's $k$-factor Theorem

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