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Spanning Eulerian subgraphs and Catlin’s reduced graphs
A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following:
(i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large;
(ii) the size of a maximum matching in G is at most 6;
(iii) the independence number of G is at most 5.
These are improvements of prior results in [16], [18], [24] and [25]
Properties of Catlin's reduced graphs and supereulerian graphs
A graph is called collapsible if for every even subset ,
there is a spanning connected subgraph of such that is the set of
vertices of odd degree in . A graph is the reduction of if it is
obtained from by contracting all the nontrivial collapsible subgraphs. A
graph is reduced if it has no nontrivial collapsible subgraphs. In this paper,
we first prove a few results on the properties of reduced graphs. As an
application, for 3-edge-connected graphs of order with for any where are given, we show how such graphs
change if they have no spanning Eulerian subgraphs when is increased from
to 10 then to
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