Graph factors modulo k

Thomassen, Carsten

doi:10.1016/j.jctb.2014.01.002

Cited by 16 publications

(25 citation statements)

References 14 publications

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“…Thomassen [18] introduced f -factors modulo k and proved that a bipartite (6k − 7)edge-connected graph has such a factor if the sum of f -values on the left-hand side is congruent to the sum of f -values on the right-hand side modulo k. In the present paper we extend this to non-bipartite graphs of sufficiently large bipartite index, in particular to graphs of large chromatic number. We apply this to the 1-2-3-conjecture using only weights 1, 2.…”

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confidence: 72%

“…In [18], the first author obtained the following result using the weak circular flow conjecture (now a theorem) [17], [9]. Theorem 2.2 (Thomassen [18]). Let k ≥ 3 be an odd integer, let G be a bipartite graph with bipartition {V 1 , V 2 }, and let f : V (G) → Z k be a mapping satisfying Equation (1).…”

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confidence: 99%

“…We shall combine this theorem with the following lemma. [16], [18]). Let q be a positive integer and let G be a (2q − 1)edge-connected graph.…”

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confidence: 99%

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The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Thomassen

Zhang

2016

Journal of Combinatorial Theory, Series B

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Let k be an odd natural number ≥ 5, and let G be a (6k − 7)-edge-connected graph of bipartite index at least k − 1. Then, for each mapping f : V (G) → N, G has a subgraph H such that each vertex v has H-degree f (v) modulo k. We apply this to prove that, if c : V (G) → Z k is a proper vertex-coloring of a graph G of chromatic number k ≥ 5 or k − 1 ≥ 6, then each edge of G can be assigned a weight 1 or 2 such that each weighted vertex-degree of G is congruent to c modulo k. Consequently, each nonbipartite (6k − 7)-edge-connected graph of chromatic number at most k (where k is any odd natural number ≥ 3) has an edge-weighting with weights 1, 2 such that neighboring vertices have distinct weighted degrees (even after reducing these weighted degrees modulo k). We characterize completely the bipartite graph having an edge-weighting with weights 1, 2 such that neighboring vertices have distinct weighted degrees. In particular, that problem belongs to P while it is NP-complete for nonbipartite graphs. The characterization also implies that every 3-edge-connected bipartite graph with at least 3 vertices has such an edge-labelling, and so does every simple bipartite graph of minimum degree at least 3.

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confidence: 72%

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confidence: 99%

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The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Thomassen

Zhang

2016

Journal of Combinatorial Theory, Series B

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“…The terminology and notation are the same as in [13] which are essentially the same as in [1], [6], [11]. In the present paper, however, a graph may have multiple edges (but no loops).…”

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confidence: 99%

Factorizing regular graphs

Thomassen

2020

Journal of Combinatorial Theory, Series B

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Every 9-regular graph (possibly with multiple edges) with odd edge-connectivity > 5 can be edge-decomposed into three 3-factors. If Tutte's 3-flow conjecture is true, it also holds for all 9-regular graphs with odd edge-connectivity 5, but not with odd edge-connectivity 3. It holds for all planar 2-edge-connected 9-regular graphs, an equivalent version of the 4-color theorem for planar graphs. We address the more general question: If G is an r-regular graph, and r = kq where k, q are natural numbers > 1, can G be edge-decomposed into k q-factors? If q is even, then the decomposition exists trivially. If k, q are both odd, then we prove that the decomposition exists if G has odd edgeconnectivity (size of smallest odd edge-cut) at least 3k − 2, which is satisfied if the odd edge-connectivity is at least r − 2. If q is odd and k is even, then we must require that G has an even number of vertices just to guarantee that G has a q-factor. If we want a decomposition into q-factors, then we also need the condition that, for any partition of the vertex set of G into two odd parts, there must be at least k edges between the parts. We prove that the edge-decomposition into q-factors is always possible if G has an even number of vertices and the edge-connectivity of G is at least 2k 2 + k.

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“…[10]). Let k be a natural number, and let G be a (3k − 2)-edge-connected bipartite graph with partition classes A and B.…”

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confidence: 99%