773 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 minimum degree conditions for supereulerian graphs
A graph is called supereulerian if it has a spanning closed trail. Let be a 2-edge-connected graph of order such that each minimal edge cut with satisfies the property that each component of has order at least . We prove that either is supereulerian or belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree : If is a 2-edge-connected graph of order with such that for every edge , we have , then either is supereulerian or belongs to one of two classes of exceptional graphs. We show that the condition cannot be relaxed
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]
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