ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 3, pp. 269–273.
Pleiades Publishing, Inc., 2011.
Original Russian Text
M.A. Babenko, T.A. Urbanovich, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 3, pp. 59–63.
Linear Algorithm for Selecting an Almost Regular
Spanning Subgraph in an Almost Regular Graph
M. A. Babenko and T. A. Urbanovich
Chair of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University
Received January 11, 2011; in ﬁnal form, February 24, 2011
Abstract—We consider almost d-regular graphs, i.e., graphs having all vertex degrees equal
to either d or d − 1. It is known that for each d
≤ d every d-regular graph contains an almost
-regular spanning subgraph. We give an algorithm for selecting such a subgraph in optimal
For an arbitrary graph G,wewriteVGand EG, respectively, to denote the set of vertices and
edges of G. An undirected graph G such that every its vertex has degree d is called d-regular.The
following question seems natural: given a d-regular graph G, what are the numbers d
there exists a d
-regular spanning subgraph G
Recall that a matching in G is a set of edges M such that every vertex v ∈ VG is incident to
at most one edge of M . The vertices incident to edges of M are said to be covered by M.Ifa
matching M covers all vertices, it is called perfect.
It is clear that for d
= 1 the problem reduces to ﬁnding a perfect matching in G. The general
case of the problem is more diﬃcult. However, it turns out possible to achieve a simple solution
for some special cases. The following statement is well known.
Theorem 1 [1, p. 344]. Every 3-regular bridgeless graph contains a perfect matching.
Note that the absence of bridges in a graph is essential.
Another important special case concerns bipartite graphs. Recall that a graph G is called
bipartite if one can partition its vertex set into two disjoint subsets (called parts ) such that the
ends of every edge of G belong to distinct parts. The next statement is a simple corollary of Hall’s
Theorem 2. Every regular bipartite graph contains a perfect matching.
The case d
> 1 can easily be reduced to d
=1. Namely,letM denote a perfect matching
in a d-regular graph G. By deleting the edges of M in G,oneobtainsa(d − 1)-regular spanning
. The latter again contains a perfect matching M
. The union M ∪ M
forms a 2-regu-
lar spanning subgraph of G. By continuing this process, one gets the following widely known
Theorem 3. Given a d-regular graph G and an integer d
≤ d, there exists a d
subgraph in G.
Apart from the mere existence of a regular subgraph, we are also interested in an eﬃcient way
for ﬁnding it. Let n := |VG| and m := |EG|. It is known  that for 3-regular bridgeless graphs