### Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

*Gabor Bacso, Zsolt Tuza*

#### Abstract

A clique-transversal set in a graph is a subset of the vertices that
meets
all maximal complete subgraphs on at least two vertices. We prove that
every
connected graph of order n and maximum degree three has a
clique-transversal
set of size ⌊ 19n/30+2/15 ⌋. This bound is
tight,
since 19n/30-1/15 is a lower bound for infinitely many
values
of n. We also prove that the vertex set of any
connected
claw-free graph of maximum degree at most four, other than an odd
cycle
longer than three, can be partitioned into two clique-transversal
sets.
The proofs of both results yield polynomial-time algorithms that find
corresponding
solutions.

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