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author:

Hou, J. (Hou, J..) [1] | Zhu, H. (Zhu, H..) [2]

Indexed by:

Scopus

Abstract:

Let k≥ 4 be an integer, and let G be a graph with maximum degree Δ. In 1991, Alon, McDiarmid and Reed proved that G has a proper coloring using O(Δ (k-1)/(k-2)) colors such that G does not have bichromatic paths with k vertices. In this paper, we improve this result by showing G has a proper coloring using (1 + ⌈ k/ 2 ⌉ 1/(k-3)) Δ (k-1)/(k-2)+ Δ + 1 colors such that G does not have bichromatic paths with k vertices. We remark that there exists a graph G with maximum degree Δ such that for any proper coloring of G using Ω(Δ(k-1)/(k-2)(logΔ)1/(k-2)) colors, there is always a bichromatic path with k vertices. Using the similar method, we also show that G has a proper coloring using O(Δ 4 / 3) colors such that G contains neither bichromatic cycles nor bichromatic paths with order 5. © 2020, Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia.

Keyword:

Acyclic; Coloring; Entropy compression; Pk-free

Community:

  • [ 1 ] [Hou, J.]Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian 350003, China
  • [ 2 ] [Zhu, H.]Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian 350003, China

Reprint 's Address:

  • [Zhu, H.]Center for Discrete Mathematics, Fuzhou UniversityChina

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Source :

Bulletin of the Malaysian Mathematical Sciences Society

ISSN: 0126-6705

Year: 2020

1 . 5 5 4

JCR@2020

1 . 0 0 0

JCR@2023

ESI HC Threshold:50

JCR Journal Grade:1

CAS Journal Grade:3

Cited Count:

WoS CC Cited Count:

SCOPUS Cited Count: 2

ESI Highly Cited Papers on the List: 0 Unfold All

WanFang Cited Count:

Chinese Cited Count:

30 Days PV: 1

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