 
Summary: PARTITIONING INTO GRAPHS WITH ONLY SMALL
COMPONENTS
NOGA ALON1
, GUOLI DING2
, BOGDAN OPOROWSKI3,4
,
AND DIRK VERTIGAN4
Abstract. The paper presents several results on edge partitions
and vertex partitions of graphs into graphs with bounded size com
ponents. We show that every graph of bounded treewidth and
bounded maximum degree admits such partitions. We also show
that an arbitrary graph of maximum degree four has a vertex par
tition into two graphs, each of which has components on at most 57
vertices. Some generalizations of the last result are also discussed.
1. Introduction
Graphs in this paper are simple, that is, without loops or multiple
edges. The set of vertices of a graph G will be denoted by V (G), and the
set of edges of G will be denoted by E(G). An edge partition of a graph
G is a set {A1, A2, . . . , Ak} of subgraphs of G such that k
i=1 E(Ai) =
