The Maximum Number of Edges of a
Spanning Eulerian Subgraph in a Graph*
Dengxin Li
College of Science,
Chongqing Technology and Business University,
Chongqing 400047, P.R.China
Abstract A graph G is supereulerian if G has a spanning eulerian subgraph. We
use SL to denote the families of supereulerian graphs. In 1995, Zhi-Hong Chen and
Hong-Jian Lai presented the following open problem [2, problem 8.8 ] : Determine
| E(H ) |
: H is a spanning eulerian subgraph of G}
GSL {K1} | E (G ) |
L  min max {
For a graph G, O(G) denotes the set of all odd-degree vertices of G. Let G be a
simple graph and| O(G) | = 2k. In this note, we show that if G  SL and k  3, then
L  2/3. By the example in [3], the condition k  3 cannot be relaxed.
Thus, if considering |O(G)| of a graph only, for the problem we have the following
result.
Theorem Let G be a supereulerian graph, and O(G) denote the set of all vertices
of G with odd degrees. Let |O(G)| = 2k. Each of the following holds.
(i)
If
k  3 and G is a simple graph, then L  2 3 .
(ii)
If
k  4 and G is a simple graph, then L  3 5 .
(iii)
If we allow G to have multiple edges, then L 
1
2
.
It is well known that if G has two edge-disjoint spanning trees, then G is
supereulerian graph [4]. We present the following conjecture to conclude this article.
Example 1:
In 1997, Lai H.J., Mao J.Z. and Li D.Y. pointed out that if multiple edges are
permitted,then
L
1
2 . In the figure 1, n  2 multiple edges are added onto an
n -cycle with n  4 .Here the maximum spanning eulerian subgraph H has n
edges and
E (G) = 2n  2 . Hence the conjecture may be valid only for simple
graphs. It is obvious that in the above example when n approaches infinite, the ratio
E(H )
n
1
1
 ; but L  .
=
2
E (G ) 2 n  2
2
Example 2:
In 2002, Li D.X. in [3] presented a counterexample to show the Catlin-conjecture is invalid
even if the graph G is simple. In the figure 2,a maximum spanning eulerian subgraph has
3m  2 edges with E (G) = 5m  2 , it follows
3m  2
3
 （ m   ）,then L  3 5
5m  2
5
figure 2 The counterexample of Catlin-conjecture
Figure 1 n=9
kkjjn=nnnnn
A property
of L
=9found a property of β:
In [5], we
Theorem let G be not a eulerian graph. If there is a maximum spanning eulerian subgraph
E(H * )
E(H )
*
=  ，a graph G will be found such that
  , where
H in G such that
E (G )
E (G * )
H * is a maximum spanning eulerian subgraph in G * .
Conjecture
Let G be a simple graph. If G has two edge-disjoint spanning trees, then
L  23 .
References
[1] J.A.Bondy and U.S.R.Murty, Graph Theory with Applications,American Esevier,
New York,1976.
[2] Zhi-Hong Chen and Hong-Jian Lai, Reduction techniques for supereulerian graphs
and related topics- An update, Combinatorics and Graph Theory 95’ Vol. World
Scientific, Singapore, (1995)53-69.
[3] Dengxin Li, Deying Li and Jingzhong Mao, On Maximum number of edges in
spaninnig eulerian subgraph, Discrete Mathematics 274 (2004)299-302.
[4] P.A.Catlin, A Reduction method to find spanning eulerian subgraphs, J.Graph
Theory 12 (1988) 29-45.
[5]李霄民，李登信，探索极大欧拉生成子图的一种方法[J]，工程数学学报，21
(2004) , 1018-1020;1036