On maximum number of edges in a spanning eulerian subgraph

The Maximum Number of Edges in a Strongly Multiplicative Graph

Spanning subgraph with Eulerian components

A 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, l...

the maximum number of edges in a strongly multiplicative graph

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A sufficient condition for the existence of a spanning Eulerian subgraph in 2-edge-connected graphs

vVe prove that if G is a 2-edge-connected graph of order n 2: 14 and max{d{u),d(v)} > n3!) for each pair of nonadjacent vertices u~ v of G. then G contains a spanning Eulerian subgraph and hence the line graph of G is Hamiltonian.

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for seve...