Orthogonal double covers of Kn,n by small graphs

El-Shanawany, R.; Gronau, Hans-Dietrich O. F.; Grüttmüller, Martin

doi:10.1016/s0166-218x(03)00269-5

Cited by 17 publications

(10 citation statements)

References 12 publications

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“…An algebraic construction of ODCs via symmetric starters has been exploited to get a complete classification of ODCs of K n,n by G for n 6 9: a few exceptions apart, all graphs G are found this way (see [4]). Much of research on this subject focused with the detection of ODCs with pages isomorphic to a given graph G. For more results on ODCs, see [2,7].…”

Section: Theorem 13 a Half-starter Is A Symmetric Starter If And Onmentioning

confidence: 99%

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Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs

El-Serafi

El-Shanawany

Shabana

2015

Ain Shams Engineering Journal

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Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with symmetric starter vectors of the orthogonal double covers (ODCs) of the complete bipartite graph and using the method of cartesian product of symmetric starter vectors to construct ODC of the complete bipartite graph by G, where G is a complete bipartite graph, disjoint union of different complete bipartite graphs and disjoint union of finite copies of a complete bipartite graph. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Section: Theorem 13 a Half-starter Is A Symmetric Starter If And Onmentioning

confidence: 99%

“…While in principle any regular graph is worth considering. The choice of H = K n,n is quite natural, also in view of a technical motivation: ODCs of such graphs are a helpful tool for constructing ODCs of K n (see [4]). …”

Section: Introductionmentioning

confidence: 99%

Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs

El-Serafi

El-Shanawany

Shabana

2015

Ain Shams Engineering Journal

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“…G is called a half starter with respect to Z n if |E(G)| = n and the lengths of all edges in G are mutually different, that is, {d(e) : e ∈ E(G)} = Z n . Theorem 2.1 (see [6]). If G is a half starter, then the set of all translates of G forms an edge decomposition of K n,n .…”

Section: Odc Of K Nn By Symmetric Startersmentioning

confidence: 99%

“…If all pages in Ᏻ are isomorphic to a graph G, then Ᏻ is said to be an ODC by G. Clearly, G must have exactly n − 1 edges. The existence of ODCs was considered by many authors [1][2][3][4][5][6][7][8][9]. Although progress on the existence problem has been made, it is far from being solved completely [4].…”

Section: Introductionmentioning

confidence: 99%

General Symmetric Starter of Orthogonal Double Covers of Complete Bipartite Graph

El-Shanawany

Higazy

2007

International Journal of Mathematics and Mathematical Sciences

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An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graphKmn,mn(for any values ofm,n) and all graphs in the collection are isomorphic to certain spanning subgraphs. Furthermore, the ODCs ofKn,nby certain disjoint stars are constructed.

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“…An algebraic construction of ODCs via "symmetric starters" (see Section 2) has been exploited to get a complete classification of ODCs of , by for ≤ 9, a few exceptions apart, all graphs are found this way (see [3, Table 1]). This method has been applied in [3,4] Much research on this subject focused on the detection of ODCs with pages isomorphic to a given graph . For a summary of results on ODCs, see [1,4].…”

Section: Introductionmentioning

confidence: 99%

On Cartesian Products of Orthogonal Double Covers

Shanawany

Higazy

El-Mesady

2013

International Journal of Mathematics and Mathematical Sciences

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Let be a graph on vertices and G a collection of subgraphs of , one for each vertex, where G is an orthogonal double cover (ODC) of if every edge of occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in and share no edges whenever the corresponding vertices are nonadjacent in . In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by new disjoint unions of complete bipartite graphs.

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Orthogonal double covers of Kn,n by small graphs

Cited by 17 publications

References 12 publications

Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs

Orthogonal double cover of Complete Bipartite Graph by disjoint union of complete bipartite graphs

General Symmetric Starter of Orthogonal Double Covers of Complete Bipartite Graph

On Cartesian Products of Orthogonal Double Covers

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