P. A. Catlin [Congr. Numerantium 76, 173–181 (1990; Zbl 0862.05058)] posed the following conjecture that is still open.
Conjecture 1. Every 3-edge-connected graph of order at most 17 has a spanning Eulerian subgraph or it can be contractible to the Petersen graph.
The second author and H.-J. Lai [Ars Comb. 48, 271–282 (1998; Zbl 0962.05035)] proved Conjecture 1 is true for those graphs of order at most 11 and even more they showed that every connected graph of order at most 11 and of minimum degree at least 3 has a reduction of order at most two of the Petersen graph. In this paper, the authors improve this result and show the following.
Theorem 2. Every 3-edge-connected graph \(G\) of order at most 15 with its reduction \(G'\) satisfies exactly one of the following conditions.

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If the order of \(G\) is at most 13, then either \(G\) has a spanning Eulerian subgraph or \(G^\prime\) is the Petersen graph.

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If the order of \(G\) is at most 14, then either \(G\) has a spanning Eulerian subgraph, or \(G^\prime\) has order 14 that can be contractible to the Petersen graph.

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If the order of \(G\) is 15, \(G\) has no spanning Eulerian subgraph and not in the graphs above, then \(G=G^\prime\) and \(G\) is both 2-connected and essentially 4-edged-connected graph with girth at least 5 and its vertex set being composed of vertices of degree 3 and 4 (of size 3 and independent).

Using Theorem 2 and some other known results on the Catlin reduction, the authors give some sufficient conditions for a 3-edge-connected graph to have a spanning Eulerian subgraph.
Theorem 3. Let \(G\) be a 3-edge-connected graph of order \(n\) satisfying exactly one of the following conditions

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the sum of degrees of two nonadjacent vertices in \(G\) is \(\frac {2n-29}{15}\),

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the size of a maximum matching of \(G\) is at most 6,

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the independence number is at most 5.

Then \(G\) has a spanning Eulerian subgraph or can be contractible to the Petersen graph.
Finally, the authors posed the following conjecture that is a refinement of Conjecture 1.
Conjecture 4. Every 3-edge-connected simple graph of order \(n\) at most 17 has a spanning Eulerian subgraph or its reduction is either the Petersen graph \(P\) or those graph obtained from \(P\) by replacing one vertex of \(P\) by two graphs, respectively.