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    Largest 2-regular subgraphs in 3-regular graphs

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    For a graph GG, let f2(G)f_2(G) denote the largest number of vertices in a 22-regular subgraph of GG. We determine the minimum of f2(G)f_2(G) over 33-regular nn-vertex simple graphs GG. To do this, we prove that every 33-regular multigraph with exactly cc cut-edges has a 22-regular subgraph that omits at most max⁑{0,⌊(cβˆ’1)/2βŒ‹}\max\{0,\lfloor (c-1)/2\rfloor\} vertices. More generally, every nn-vertex multigraph with maximum degree 33 and mm edges has a 22-regular subgraph that omits at most max⁑{0,⌊(3nβˆ’2m+cβˆ’1)/2βŒ‹}\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\} vertices. These bounds are sharp; we describe the extremal multigraphs
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