Volume 16, No 2, 2009, P. 1620
UDC 519.172.2
O. V. Borodin, A. O. Ivanova
Nearproper vertex 2colorings of sparse graphs
Abstract:
A graph $G$ is $(2,1)$colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ any component contains at most two vertices while $G[V_2]$ is edgeless. We prove that every graph $G$ with the maximum average degree $\mathrm{mad}(G)$ smaller than 7/3 is $(2,1)$colorable. It follows that every planar graph with girth at least 14 is $(2,1)$colorable. We also construct a planar graph $G_n$ with $\mathrm{mad}(G_n)=(18n2)/(7n1)$ that is not $(2,1)$colorable.
Bibl. 5.
Keywords: planar graph, girth, coloring, partition.
Borodin Oleg Veniaminovich ^{1}
Ivanova Anna Olegovna ^{2}
1. S. L. Sobolev Institute of Mathematics, SB RAS,
4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
2.
Institute of Mathematics at Yakutsk State University,
677891 Yakutsk, Russia
email: brdnoleg@math.nsc.ru, shmgnanna@mail.ru
