Gallai's Path Decomposition for 2-degenerate Graphs
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Gallai's path decomposition conjecture states that if G is a connected graph on n vertices, then the edges of G can be decomposed into at most ⌈n /2⌉ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on 2k+1 vertices by deleting at most k-1 edges. Bonamy and Perrett asked if every connected graph G on n vertices can be decomposed into at most ⌊n/2⌋ paths unless G is an odd semi-clique. A graph G is said to be 2-degenerate if every subgraph of G has a vertex of degree at most 2. In this paper, we prove that any connected 2-degenerate graph G on n vertices can be decomposed into ⌊n /2⌋ paths unless G is a triangle.
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