 
Summary: Combinatorics, Probability and Computing (1993) 11, 110
Copyright c 1993 Cambridge University Press
The chromatic number of graph powers
NOGA ALON1
and BO JA N MO H A R2
1 Department of Mathematics, Tel Aviv University
Tel Aviv, Israel
2 Department of Mathematics, University of Ljubljana
1111 Ljubljana, Slovenia
Received
It is shown that the maximum possible chromatic number of the square of a graph
with maximum degree d and girth g is (1+o(1))d2 if g = 3, 4, 5 or 6, and is (d2/ log d)
if g 7. Extensions to higher powers are considered as well.
1. Introduction
The square G2
of a graph G = (V, E) is the graph whose vertex set is V and in which two
distinct vertices are adjacent if and only if their distance in G is at most 2. What is the
maximum possible chromatic number of G2
, as G ranges over all graphs with maximum
degree d and girth g?
