Approximating the chromatic index of multigraphs

Chen, Guantao; Yu, Xingxing; Zang, Wenan

doi:10.1007/s10878-009-9232-y

Cited by 17 publications

(24 citation statements)

References 19 publications

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“…This technical fact was used in several results using Tashkinov trees, for example [1,5,11]. Our main lemma, Lemma 8, is also based on this parameter.…”

Section: Toolsmentioning

confidence: 99%

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Edge coloring multigraphs without small dense subsets

Haxell

Kierstead

2015

Discrete Mathematics

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a b s t r a c tOne consequence of a long-standing conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose G is a multigraph with maximum degree ∆, such that no vertex subset S of odd size at most ∆ induces more than (∆+1)(|S|−1)/2 edges. Then G has an edge coloring with ∆+1 colors.Here we prove a weakened version of this statement.

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“…This technical fact was used in several results using Tashkinov trees, for example [1,5,11]. Our main lemma, Lemma 8, is also based on this parameter.…”

Section: Toolsmentioning

confidence: 99%

“…Here we mention just the results that directly relate to this note. The best known approximate version is as follows, due to Scheide [11] (independently proved by Chen, Yu and Zang [1], see also [12] and [2]), who proved that the conjecture is true when ⌈ρ(G)⌉ ≥ ∆ +  ∆−1 2 .…”

Section: Conjecture 2 For Every Multigraphmentioning

confidence: 99%

Edge coloring multigraphs without small dense subsets

Haxell

Kierstead

2015

Discrete Mathematics

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“…We remark that many of the results and ideas used here have appeared in other works (e.g. [6,26,30,31]) but in order to keep this paper self-contained we include all proofs. Lemma 2.8.…”

Section: Toolsmentioning

confidence: 99%

“…In the past few years, there were several improvements for the upper bound on χ ′ (G) in terms of ρ(G), see, e.g., [6,14,15,16,24,26,27,31]. In [6], and independently [26], it was shown that χ ′ (G) ≤ max{∆+ ∆ 2 , ⌈ρ(G)⌉}. A very recent breakthrough by Chen, Gao, Kim, Postle, and Shan [5] proves the best known upper bound of χ ′ (G) ≤ max{∆ + 3 ∆ 2 , ⌈ρ(G)⌉}.…”

Section: Introductionmentioning

confidence: 99%

Goldberg's conjecture is true for random multigraphs

Haxell

Krivelevich

Kronenberg

2019

Journal of Combinatorial Theory, Series B

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In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph G,We show that their conjecture (in a stronger form) is true for random multigraphs. Let M (n, m) be the probability space consisting of all loopless multigraphs with n vertices and m edges, in which m pairs from [n] are chosen independently at random with repetitions. Our result states that, for a given m := m(n), M ∼ M (n, m) typically satisfies χ ′ (G) = max{∆(G), ⌈ρ(G)⌉}. In particular, we show that if n is even and m := m(n), then χ ′ (M ) = ∆(M ) for a typical M ∼ M (n, m). Furthermore, for a fixed ε > 0, if n is odd, then a typical M ∼ M (n, m) has χ ′ (M ) = ∆(M ) for m ≤ (1 − ε)n 3 log n, and χ ′ (M ) = ⌈ρ(M )⌉ for m ≥ (1 + ε)n 3 log n. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.

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“…Feige et al [12] show a polynomial-time algorithm which, for a given multigraph and an integer k, finds a subgraph H such that |E(H)| ≥ OPT, ∆(H) ≤ k + 1 and Γ(H) ≤ k+ √ k + 1+2, where OPT is the number of edges in the maximum k-edge colorable sugraph of G, and Γ(H) is the odd density of H, defined as Γ(H) = max S⊆V (H),|S|≥2 |E(S)| ⌊|S|/2⌋ . The subgraph H can be edge-colored with at most max{∆+ ∆/2, ⌈Γ(H)⌉} ≤ ⌈k + √ k + 1+2⌉ colors in n O( √ k) -time by an algorithm of Chen, Yu and Zang [3]. By choosing the k largest color classes as a solution this gives a k/⌈k + √ k + 1 + 2⌉-approximation.…”

Section: Approximation Algorithms For the Max K-ecs Problemmentioning

confidence: 99%

Beyond the Vizing's Bound for at Most Seven Colors

Kamiński

Kowalik

2014

SIAM J. Discrete Math.

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Let G = (V, E) be a simple graph of maximum degree Δ. The edges of G can be colored with at most Δ + 1 colors by Vizing's theorem. We study lower bounds on the size of subgraphs of G that can be colored with Δ colors. Vizing's theorem gives a bound of Δ Δ+1 |E|. This is known to be tight for cliques K Δ+1 when Δ is even. However, for Δ = 3 it was improved to 26 31 |E| by Albertson and Haas [Discrete Math., 148 (1996), pp. 1-7] and later to 6 7 |E| by Rizzi [Discrete Math., 309 (2009), pp. 4166-4170]. It is tight for B 3 , the graph isomorphic to a K 4 with one edge subdivided. We improve previously known bounds for Δ ∈ {3, . . . , 7}, under the assumption that for Δ = 3, 4, 6, graph G is not isomorphic to B 3 , K 5 , and K 7 , respectively. For Δ ≥ 4 these are the first results which improve over the Vizing's bound. We also show a new bound for subcubic multigraphs not isomorphic to K 3 with one edge doubled. In the second part, we give approximation algorithms for the maximum k-edge-colorable subgraph problem, where given a graph G (without any bound on its maximum degree or other restrictions) one has to find a k-edge-colorable subgraph with the maximum number of edges. In particular, when G is simple for k = 3, 4, 5, 6, 7 we obtain approximation ratios of 13 15 , 9 11 , 19 22 , 23 27 , and 22 25 , respectively. We also present a 7 9 -approximation for k = 3 when G is a multigraph. The approximation algorithms follow from a new general framework that can be used for any value of k.

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Approximating the chromatic index of multigraphs

Abstract: It is well known that if G is a multigraph then χ (G) ≥ χ

Cited by 17 publications

References 19 publications

Edge coloring multigraphs without small dense subsets

Edge coloring multigraphs without small dense subsets

Goldberg's conjecture is true for random multigraphs

Beyond the Vizing's Bound for at Most Seven Colors

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