Existence of 3-regular subgraphs in Cartesian product of cycles

Borse, Y. M.; Saraf, J. B.

doi:10.1016/j.akcej.2018.07.001

Cited by 6 publications

(5 citation statements)

References 8 publications

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“…Borse and Shaikh [2] improved this result by proving that for such values of l there exists a 3-regular subgraph of Q n with l vertices which is 3-connected and bipancyclic too. Similar results for the classes of the Cartesian product of cycles and the Cartesian product of paths are obtained in [1] and [13], respectively. For the existence of 4-regular subgraphs, Borse and Shaikh [3] established that there exists a 4-regular, 4-connected and bipancyclic subgraph on l vertices in the hypercube Q n if and only if l ¼ 16 or l is an even integer with 24 l 2 n :…”

Section: Introductionsupporting

confidence: 76%

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On regular subgraphs of augmented cubes

Shinde

Borse

2020

AKCE International Journal of Graphs and Combinatorics

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The n-dimensional augmented cube AQ n is a variation of the hypercube Q n : It is a ð2n À 1Þ-regular and ð2n À 1Þ-connected graph on 2 n vertices. One of the fundamental properties of AQ n is that it is pancyclic, that is, it contains a cycle of every length from 3 to 2 n : In this paper, we generalize this property to k-regular subgraphs for k ¼ 3 and k ¼ 4: We prove that the augmented cube AQ n with n ! 4 contains a 4-regular, 4-connected and pancyclic subgraph on l vertices if and only if 8 l 2 n : Also, we establish that for every even integer l from 4 to 2 n , there exists a 3-regular, 3-connected and pancyclic subgraph of AQ n on l vertices.

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Section: Introductionsupporting

confidence: 76%

“…Write AQ 4 ¼ AQ 0 3 [ AQ 1 3 [ E h [ E c: Suppose m 2 f4, 5, 6g: Then there exists an m-cycle C containing a path ha 1 , a 2 , a 3 i in AQ 0 3 and a vertex v 2 VðAQ 0 3 Þ À VðCÞ such that (i) v is adjacent to a 1 , a 2 , a 3 ; (ii) v is adjacent to the vertex b 3 of AQ1 3 corresponding to a 3 :…”

mentioning

confidence: 99%

On regular subgraphs of augmented cubes

Shinde

Borse

2020

AKCE International Journal of Graphs and Combinatorics

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“…Taken this together with Proposition 4.7 and the equivalence of Condition (1), ( 2) and (3), Condition (1) implies Condition (4) and (5). Condition (4) and ( 5) taken together with Equation (a) imply Condition (2) and (3), respectively.…”

Section: -Well-behavedmentioning

confidence: 70%

“…finding so-called 1-factorizations, determining so-called preclusion numbers or enumeration of perfect k-matchings. 3-matchings in the Cartesian product of cycles have been studied in [5] and k-matchings of hypercubes (the Cartesian product of edges) in [28]. In particular, it seems to be a non-trivial endeavor to characterize k-matchings of general graph products with arbitrary factors in terms that are solely based on the structure of these factors.…”

Section: Introductionmentioning

confidence: 99%

Construction of k-matchings in graph products

Lindeberg

Hellmuth

2022

Art Discrete Appl. Math.

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A k-matching M of a graph G = (V, E) is a subset M ⊆ E such that each connected component in the subgraph F = (V, M ) of G is either a single-vertex graph or k-regular, i.e., each vertex has degree k. In this contribution, we are interested in k-matchings in the four standard graph products: the Cartesian, strong, direct and lexicographic product.As we shall see, the problem of finding non-empty k-matchings (k ≥ 3) in graph products is NP-complete. Due to the general intractability of this problem, we focus on different polynomial-time constructions of k-matchings in a graph product G ⋆ H that are based on k G -matchings M G and k H -matchings M H of its factors G and H, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum k-matching in the respective products. Such constructions are also called "well-behaved" and we provide several characterizations for this type of k-matchings.Our specific constructions of k-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never "projected" to unmatched edges in the factors. This leads to the concept of weak-homomorphism preserving k-matchings. Although the specific k-matchings constructed here are not always maximum k-matchings of the products, they have always maximum size among all weak-homomorphism preserving k-matchings. Not all weakhomomorphism preserving k-matchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphism preserving k-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

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“…finding so-called 1-factorizations, determining so-called preclusion numbers or enumeration of perfect k-matchings. 3-matchings in the Cartesian product of cycles have been studied in [6] and k-matchings of hypercubes (the Cartesian product of edges) in [28]. In particular, it seems to be a non-trivial endeavour to characterize k-matchings of general graph products with arbitrary factors in terms that are solely based on the structure of these factors.…”

Section: Introductionmentioning

confidence: 99%

Construction of $k$-matchings and $k$-regular subgraphs in graph products

Lindeberg¹,

Hellmuth²

2021

Preprint

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A k-matching M of a graph G = (V, E) is a subset M ⊆ E such that each connected component in the subgraph F = (V, M ) of G is either a single-vertex graph or k-regular, i.e., each vertex has degree k. In this contribution, we are interested in k-matchings within the four standard graph products: the Cartesian, strong, direct and lexicographic product.As we shall see, the problem of finding non-empty k-matchings (k ≥ 3) in graph products is NP-complete. Due to the general intractability of this problem, we focus on distinct polynomial-time constructions of k-matchings in a graph product G H that are based on k Gmatchings M G and k H -matchings M H of its factors G and H, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum k-matching in the respective products. Such constructions are also called "well-behaved" and we provide several characterizations for this type of k-matchings.Our specific constructions of k-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never "projected" to unmatched edges in the factors. This leads to the concept of weakhomomorphism preserving k-matchings. Although the specific k-matchings constructed here are not always maximum k-matchings of the products, they have always maximum size among all weak-homomorphism preserving k-matchings. Not all weak-homomorphism preserving kmatchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphims preserving k-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

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Existence of 3-regular subgraphs in Cartesian product of cycles

Cited by 6 publications

References 8 publications

On regular subgraphs of augmented cubes

On regular subgraphs of augmented cubes

Construction of k-matchings in graph products

Construction of $k$-matchings and $k$-regular subgraphs in graph products

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