# Hamilton decompositions of cycle plus triangles graphs

A cycle plus triangles graph is a 4-regular graph $$G$$ with a Hamiltonian circuit $$C$$ and such that the chords of $$C$$ induce a set of disjoint triangles (3-circuits). A 4-regular graph $$G$$ has a Hamilton decomposition if its set of edges can be partitioned into two sets that induce Hamiltonian circuits in $$G$$. A cycle plus triangles graph is edge 3-connected if it doesn’t have any edge 2-cuts (sets of two edges which when removed disconnect the graph).

Our question is the following

Question Let $$G$$ be a cycle plus triangles graph which is edge 3-connected. Is it true then that $$G$$ has a Hamiltonian decomposition, or is there a counterexample that might be known?

• As a minor comment, a 4-regular graph which is 3-edge-connected is automatically 4-edge-connected: by counting odd-degree vertices, there cannot be a partition $V(G) = S \cup T$ with exactly 3 (or any other odd number) of edges between $S$ and $T$. – Misha Lavrov Nov 24 '19 at 17:10
• @MishaLavrov Thank you! – EGME Nov 24 '19 at 17:33
• @MishaLavrov I couldn’t find cases of even order which are class 2 yet. – EGME Nov 27 '19 at 5:28
• @MishaLavrov I now checked the example in the answer – EGME Nov 27 '19 at 7:33