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

Luo, Rong; Zang, Wenan; Zhang, Cun-Quan

doi:10.1007/s00493-004-0039-2

Cited by 19 publications

(10 citation statements)

References 10 publications

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“…(• means the cell is empty.) In fact, T is the Latin bitrade corresponding to a 2-simultaneous edge coloring of the graph G that showed in Figure 1 Since Luo et al in [11] showed that each bipartite graphic sequence S with all its elements greater than 1, has a 2-simultaneous edge colorable bipartite realization, we have…”

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

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

confidence: 99%

“…Also, they conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al in [11] showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 4-NZF. Thus, by Theorems A (iii) and B, they proved that the SE conjecture is true.…”

Section: Introductionmentioning

confidence: 99%

On the simultaneous edge coloring of graphs

Gh.

Omoomi

2014

Discrete Math. Algorithm. Appl.

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“…Jaeger [2] proved that every 4-edge connected graph has a 4-NZF. In [3], the authors showed that if every edge of a graph G is contained in a cycle of length at most 4, then G admits a 4-NZF. In this paper, we determine the exact minimum flow number of G ∨ H for all graphs G and H.…”

Section: Introductionmentioning

confidence: 99%

Join of two graphs admits a nowhere-zero 3-flow

et al. 2014

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“…It has been extensively studied whether a degree sequence has a realization with certain properties. A noticeable application (see [64]) of graph realization with 4-flows has been found in the design of critical partial Latin squares which leads to the proof of the socalled simultaneous edge-coloring conjecture by Keedwell [41] and Cameron [14]. All graphic sequences which have realizations admitting a nowhere-zero 3-flow or 4-flow are characterized in [63] and [64], respectively.…”

Section: Graphic Degree Sequences With Z K -Connected Realizationsmentioning

confidence: 99%

Group Connectivity and Modulo Orientations of Graphs

Li¹

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This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte's 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte's 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowherezero 3-flow. Jaeger et al. (1992) further conjectured that every 5-edge-connected graph is Z 3-connected, whose truth implies the 3-Flow Conjecture. Extending Tutte's flows conjectures, Jaeger's Circular Flow Conjecture (1981) says every 4p-edge-connected graph admits a modulo (2p + 1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo 2p + 1 at every vertex. Note that the p = 1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of p = 2 implies the 5-Flow Conjecture. This work is devoted to provide some partial results on these problems. It is proved in Chapter 2 that every graph with four edge-disjoint spanning trees is Z 3connected. Consequently, Jaeger et al.'s group connectivity conjecture and Tutte's 3-Flow Conjecture hold for 5-edge-connected essentially 23-edge-connected graphs. We also provide several equivalent versions of Jaeger et al.'s group connectivity conjecture and indicate that it is enough to verify the conjecture for 5-edge-connected essentially 8-edge-connected graphs. In Chapter 3, Tutte's 3-Flow Conjecture is verified for graphs with independence number at most 4. The relation of orientation and group connectivity is studied in Chapter 4. It shows that every strongly Z m-connected graph contains m − 1 edge-disjoint spanning trees, and hence every Z m-connected graph G has (m − 1)(|V (G)| − 1)/(m − 2) edges, which solves a conjecture of Luo et al. (2012). Those results are applied to establish some monotonicity properties of group connectivity that every strongly Z 5-connected graph is Z 3-connected, and every Z 3-connected graph is A-connected for any Abelian group A with size |A| ≥ 4. Infinite families of counterexamples to Jaeger's Circular Flow Conjecture are presented in Chapter 5. For p ≥ 3, there are 4p-edge-connected graphs not admitting modulo (2p + 1)orientation; for p ≥ 5, there are (4p + 1)-edge-connected graphs not admitting modulo (2p + 1)orientation. Towards the p = 2 case of Circular Flow Conjecture and the 5-Flow Conjecture, we show in Chapter 6 that every 10-edge-connected planar graph admits a modulo 5-orientation.

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

Cited by 19 publications

References 10 publications

On the simultaneous edge coloring of graphs

On the simultaneous edge coloring of graphs

Join of two graphs admits a nowhere-zero 3-flow

Group Connectivity and Modulo Orientations of Graphs

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