Minimum 2-distance coloring of planar graphs and channel assignment

Zhu, Junlei; Bu, Yuehua

doi:10.1007/s10878-018-0285-7

Cited by 16 publications

(4 citation statements)

References 11 publications

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“…Observe that by the already mentioned result by Zhu and Bu [20] we can safely assume that ∆ ≥ 6. Furthermore, we can assume that G is connected, as the coloring of G can be obtained by coloring each connected component independently and each of them is smaller than G.…”

Section: Main Proofmentioning

confidence: 95%

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Coloring squares of planar graphs with small maximum degree

Mateusz¹,

Rzążewski²,

Szymon³

2021

Preprint

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For a graph G, by χ 2 (G) we denote the minimum integer k, such that there is a kcoloring of the vertices of G in which vertices at distance at most 2 receive distinct colors. Equivalently, χ 2 (G) is the chromatic number of the square of G. In 1977 Wegner conjectured that if G is planar and has maximum degree, and 3∆/2 + 1 if ∆ ≥ 8. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of ∆. In this work we show that for every planar graph G with maximum degree ∆ it holds that χ 2 (G) ≤ 3∆ + 4. This result provides the best known upper bound for 6 ≤ ∆ ≤ 14.

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Section: Main Proofmentioning

confidence: 95%

“…Every planar graph G with maximum degree ∆ satisfies χ 2 (G) ≤ 3∆ + 4. Maximum degree ∆ Conjectured bound [18] Thomassen [16] Zhu and Bu [20] Wong [19] Madaras and Marcinova [12] Borodin et al [5] Theorem 1 We point out that Theorem 1 provides the best known upper bound for the cases 6 ≤ ∆ ≤ 14.…”

Section: Molloy and Salavatipourmentioning

confidence: 99%

Coloring squares of planar graphs with small maximum degree

Mateusz¹,

Rzążewski²,

Szymon³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Conjectured bound [22] Thomassen [20] Zhu and Bu [24] Bousquet et al [7] Wong [23] Madaras and Marcinova [16] Borodin et al [6] Theorem 1 Theorem 1. Every planar graph G with maximum degree ∆ ≥ 6 satisfies χ 2 (G) ≤ 3∆ + 4.…”

Section: Agnarsson and Halldórssonmentioning

confidence: 99%

Coloring squares of planar graphs with small maximum degree

Mateusz¹,

Rzążewski²,

Szymon³

2024

Discuss. Math. Graph Theory

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For a graph G, by χ 2 (G) we denote the minimum integer k, such that there is a k-coloring of the vertices of G in which vertices at distance at most 2 receive distinct colors. Equivalently, χ 2 (G) is the chromatic number of the square of G. In 1977 Wegner conjectured that if G is planar and has maximum degree ∆, then χ 2 (G) ≤ 7 if ∆ ≤ 3, χ 2 (G) ≤ ∆ + 5 if 4 ≤ ∆ ≤ 7, and 3∆/2 + 1 if ∆ ≥ 8. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of ∆. In this work we show that for every planar graph G with maximum degree ∆ it holds that χ 2 (G) ≤ 3∆ + 4. This result provides the best known upper bound for 6 ≤ ∆ ≤ 14.

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“…∆ 749 χ 2 (G) ⌊ 9∆ 5 ⌋ + 2 Molloy and Salavatipour [10] ∆ 249 χ 2 (G) ⌈ 5∆ 3 ⌉ + 25 χ 2 (G) 5∆ 3 ⌉ + 78 Zhu and Bu [15] ∆ 5…”

Section: Agnarsson and Halldorsson [1]mentioning

confidence: 99%

Improved square coloring of planar graphs

Bousquet¹,

Deschamps²,

Meyer³

et al. 2021

Preprint

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Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that 3 2 ∆ + 1 colors are sufficient to square color every planar graph of maximum degree ∆. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that 2∆ + 7 colors are sufficient, which improves the best known bounds when 6 ∆ 31.

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Minimum 2-distance coloring of planar graphs and channel assignment

Cited by 16 publications

References 11 publications

Coloring squares of planar graphs with small maximum degree

Coloring squares of planar graphs with small maximum degree

Coloring squares of planar graphs with small maximum degree

Improved square coloring of planar graphs

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