On a conjecture of Keedwell and the cycle double cover conjecture

Mahdian, Mohammad; Mahmoodian, E. S.; Saberi, Amin; Salavatipour, Mohammad R.; Tusserkani, Ruzbeh

doi:10.1016/s0012-365x(99)00356-8

Cited by 6 publications

(13 citation statements)

References 2 publications

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“…Theorem D. [13] For every bipartite graphic sequence S with all its elements greater than one, there exists a 2-edge-connected realization. Theorem 1.…”

Section: µ-Simultaneous Edge Coloring and µ-Way Latin Tradementioning

confidence: 99%

“…A sequence S of positive integers is called a bipartite graphic sequence if there exists a bipartite graph G whose bipartite degree sequence is S; if so then the graph G is called a realization of S. Definition 1. [13] A µ-simultaneous edge coloring of graph G is a set of µ proper edge colorings of G with the color set [l], say (c 1 , c 2 , . .…”

Section: Introductionmentioning

confidence: 99%

“…The µ-simultaneous edge coloring of bipartite graphs has a close relation with µ-way Latin trades. Mahdian et al (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring.…”

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confidence: 99%

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On the simultaneous edge coloring of graphs

Gh.

Omoomi

2014

Discrete Math. Algorithm. Appl.

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A µ-simultaneous edge coloring of graph G is a set of µ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The µ-simultaneous edge coloring of bipartite graphs has a close relation with µ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the µ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extermal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

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“…Theorem D. [13] For every bipartite graphic sequence S with all its elements greater than one, there exists a 2-edge-connected realization. Theorem 1.…”

Section: µ-Simultaneous Edge Coloring and µ-Way Latin Tradementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the simultaneous edge coloring of graphs

Gh.

Omoomi

2014

Discrete Math. Algorithm. Appl.

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“…Keedwell's conjecture is shown to be true for almost all 2-way latin trades (see [16] and [10]). But as we shall see in Theorem 2.2, a similar statement is not necessarily true for any µ ≥ 3.…”

Section: Introductionmentioning

confidence: 95%

On the possible volumes of μ -way latin trades

Adams

Billington

Bryant

et al. 2002

Aequationes Mathematicae

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A µ-way latin trade of volume s is a set of µ partial latin rectangles (of inconsequential size) containing exactly the same s filled cells, such that if cell (i, j) is filled, it contains a different entry in each of the µ partial latin rectangles, and such that row i in each of the µ partial latin rectangles contains, set-wise, the same symbols and column j, likewise. In this paper we show that all µ-way latin trades with sufficiently large volumes exist, and state some theorems on the non-existence of µ-way latin trades of certain volumes. We also find the set of possible volumes (that is, the volume spectrum) of µ-way latin trades for µ = 4 and 5. (The case µ = 2 was dealt with by Fu, and the case µ = 3 by the present authors.) (2000). 05B15. Mathematics Subject Classification

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“…A graph theory version of the conjecture (as described in Theorem 1.2) was reproposed by Cameron [2]. Theorem 1.1 was proved by Hajiaghaee et al [7] for δ ≥ 4 and by Keedwell [5], Mahdian et al [9] for some other special cases.…”

Section: Introductionmentioning

confidence: 99%

Nowhere-Zero 4-Flows, Simultaneous Edge-Colorings, And Critical Partial Latin Squares

2004

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It is proved in this paper that every bipartite graphic sequence with the minimum degree δ ≥ 2 has a realization that admits a nowhere-zero 4-flow. This result implies a conjecture originally proposed by Keedwell (1993) and reproposed by Cameron (1999) about simultaneous edge-colorings and critical partial Latin squares.A corollary of Theorem 1.1 solves a conjecture originally proposed by Keedwell [5] and reproposed by Cameron [2]. Theorem 1.2. (Keedwell-Cameron Conjecture)Every bipartite graphic sequence S with the minimum degree δ(S) ≥ 2 has a realization G of S such that G has two proper edge-colorings with the following properties:(1) for any vertex, the set of colors appearing on edges at that vertex are the same in both colorings;(2) no edge receives the same color in both colorings.

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On a conjecture of Keedwell and the cycle double cover conjecture

Cited by 6 publications

References 2 publications

On the simultaneous edge coloring of graphs

On the simultaneous edge coloring of graphs

On the possible volumes of μ -way latin trades

Nowhere-Zero 4-Flows, Simultaneous Edge-Colorings, And Critical Partial Latin Squares

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