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Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into
$n / (d+1)$
paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when
$d = \Omega(n)$
, improving a result of Han, who showed that in this range almost all vertices of G can be covered by
$n / (d+1) + 1$
vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if
$d = \Omega(n)$
and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).
We consider the problem of minimizing the number of edges that are contained in triangles, among n-vertex graphs with a given number of edges. For sufficiently large n, we prove an exact formula for this minimum, which partially resolves a conjecture of Füredi and Maleki.
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