1,114 research outputs found
Spanning subgraph with Eulerian components
AbstractA graph is k-supereulerian if it has a spanning even subgraph with at most k components. We show that if G is a connected graph and G′ is the (collapsible) reduction of G, then G is k-supereulerian if and only if G′ is k-supereulerian. This extends Catlin’s reduction theorem in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. For a graph G, let F(G) be the minimum number of edges whose addition to G create a spanning supergraph containing two edge-disjoint spanning trees. We prove that if G is a connected graph with F(G)≤k, where k is a positive integer, then either G is k-supereulerian or G can be contracted to a tree of order k+1. This is a best possible result which extends another theorem of Catlin, in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. Finally, we use these results to give a sufficient condition on the minimum degree for a graph G to bek-supereulerian
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
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
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]
Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II
Let be a connected graph with and with the spanning
forest . Let be a real number and let be a real function. In this paper, we show that if for all
, , then has a spanning tree
containing such that for each vertex , , where
denotes the number of components of and denotes the
number of edges of with both ends in . This is an improvement of several
results and the condition is best possible. Next, we also investigate an
extension for this result and deduce that every -edge-connected graph
has a spanning subgraph containing edge-disjoint spanning trees such
that for each vertex , , where ; also if contains
edge-disjoint spanning trees, then can be found such that for each vertex
, , where .
Finally, we show that strongly -tough graphs, including -tough
graphs of order at least three, have spanning Eulerian subgraphs whose degrees
lie in the set . In addition, we show that every -tough graph has
spanning closed walk meeting each vertex at most times and prove a
long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed
walk; connected factor; toughness; total exces
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