Distance Constraints on Short Cycles for 3-Colorability of Planar graphs

Kang, Yingli; Wang, Yingqian

doi:10.1007/s00373-014-1476-3

Cited by 5 publications

(1 citation statement)

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“…Lemma 3.8 also implies that not both f 1 and f 2 are 2-ceiling 5-faces. So, v sends to f 1 and f 2 a total charge at most 13 6 + 3 2 , giving ch * (v) ≥ ch(v) + 3 − 13 6 − 3 2 = 1 3 > 0. Case 1.2: suppose d(v) ≥ 4. v sends charge out, only by R12, possibly to ceiling 3-or 5-or 7-faces, sticking 3-or 5-faces and pendent 3-faces.…”

Section: Structural Properties Of the Minimal Counterexample Gmentioning

confidence: 99%

(1,0,0)-colorability of planar graphs without cycles of length 4 or 6

Wang

Yang

2020

Preprint

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A graph G is (d1, d2, d3)-colorable if the vertex set V (G) can be partitioned into three subsets V1, V2 and V3 such that for i ∈ {1, 2, 3}, the induced graph G[Vi] has maximum vertex-degree at most di. So, (0, 0, 0)-colorability is exactly 3-colorability.The well-known Steinberg's conjecture states that every planar graph without cycles of length 4 or 5 is 3-colorable. As this conjecture being disproved by Cohen-Addad etc. in 2017, a similar question, whether every planar graph without cycles of length 4 or i is 3-colorable for a given i ∈ {6, . . . , 9}, is gaining more and more interest. In this paper, we consider this question for the case i = 6 from the viewpoint of improper colorings. More precisely, we prove that every planar graph without cycles of length 4 or 6 is (1,0,0)-colorable, which improves on earlier results that they are (2,0,0)-colorable and also (1,1,0)-colorable, and on the result that planar graphs without cycles of length from 4 to 6 are (1,0,0)-colorable.

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Section: Structural Properties Of the Minimal Counterexample Gmentioning

confidence: 99%