Explanation:
To determine if a connected graph is eulerian, we need to consider the degrees of its vertices. The degree of a vertex is the number of edges incident to it.

A connected graph is eulerian if and only if every vertex has an even degree. This is because, in an eulerian graph, we can trace a path that visits every edge exactly once and returns to the starting vertex. If a vertex has an odd degree, the path will not be able to return to that vertex, and therefore the graph cannot be eulerian.

We are given four combinations of vertex degrees: a) 1, 2, 3; b) 2, 3, 4; c) 2, 4, 5; d) 1, 3, 5.

Analysis of each combination:
a) 1, 2, 3:
- The graph has three vertices with degrees 1, 2, and 3.
- The vertex with degree 1 cannot be part of an eulerian path since it will not be able to return to itself.
- The remaining two vertices have even degrees and can be part of an eulerian path.
- Therefore, the graph is not eulerian.

b) 2, 3, 4:
- The graph has three vertices with degrees 2, 3, and 4.
- All three vertices have even degrees.
- Therefore, the graph is eulerian.

c) 2, 4, 5:
- The graph has three vertices with degrees 2, 4, and 5.
- The vertex with degree 5 cannot be part of an eulerian path since it will not be able to return to itself.
- The remaining two vertices have even degrees and can be part of an eulerian path.
- Therefore, the graph is not eulerian.

d) 1, 3, 5:
- The graph has three vertices with degrees 1, 3, and 5.
- The vertex with degree 1 cannot be part of an eulerian path since it will not be able to return to itself.
- The remaining two vertices have odd degrees and cannot be part of an eulerian path.
- Therefore, the graph is not eulerian.

Conclusion:
From the analysis above, we can conclude that the only combination of degrees that would result in an eulerian graph is b) 2, 3, 4.