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A graph G = (V (G),E(G)) is supereulerian if it has a spanning Eulerian subgraph. Let l(G) be the maximum number of edges of spanning Eulerian subgraphs of a graph G. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if G is an r-regular supereulerian graph, then
l(G) \geq 2/3|E(G)| when r \neq 5, and l(G) > 3/ 5|E(G)| when
r = 5. In this paper we improve the coefficient and prove that if G is a 5-regular supereulerian graph, then l(G) \geq 19/ 30|E(G)|+ 4/ 3. For this, we first show that each graph G with maximum degree at most 3 has a matching with at least
2/ 7|E(G)| edges and this bound is sharp. Moreover, we show that Catlin's conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin's conjecture does not hold for claw-free graphs in general.
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