TY - JOUR

T1 - On the weak 2-coloring number of planar graphs

AU - Almulhim, Ahlam

AU - Kierstead, H. A.

N1 - Funding Information:
The authors would like to thank and express appreciation to the Deanship of Scientific Research at King Faisal University , Saudi Arabia, for the financial support under Nasher 2021, Grant No. 216058 .
Publisher Copyright:
© 2021 The Authors

PY - 2022/1

Y1 - 2022/1

N2 - For a graph G=(V,E), a total ordering L on V, and a vertex v∈V, let Wcol2[G,L,v] be the set of vertices w∈V for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number wcol2(G) of G is the least k such that there is a total ordering L on V with |Wcol2[G,L,v]|≤k for all vertices v∈V. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.

AB - For a graph G=(V,E), a total ordering L on V, and a vertex v∈V, let Wcol2[G,L,v] be the set of vertices w∈V for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number wcol2(G) of G is the least k such that there is a total ordering L on V with |Wcol2[G,L,v]|≤k for all vertices v∈V. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.

KW - Planar graph

KW - Star chromatic number

KW - Weak 2-coloring number

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U2 - 10.1016/j.disc.2021.112631

DO - 10.1016/j.disc.2021.112631

M3 - Article

AN - SCOPUS:85116398321

SN - 0012-365X

VL - 345

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

M1 - 112631

ER -