Spanning cycles of nearly cubic graphs

AbstractThe fact that a cubic hamiltonian graph must have at least three spanning cycles suggests the question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles. We answer this in the negative by exhibiting graphs on n=2m+1, m≥5, points in which one point has degree 4, all others have degree 3, and only two spanning cycles exist

The number of cutvertices in graphs with given minimum degree

AbstractThe maximum number of cutvertices in a connected graph of order n having minimum degree at least δ is determined for δ⩾5

Matrices permutable to ∗ matrices

AbstractA (0, 1) matrix (aij) is said to be a ∗ matrix iff aij = 1 implies ai′j′ = 1 for all (i′, j′) satisfying 1 ⩽ i′ ⩽ i, 1 ⩽ j′ ⩽ j. (0, 1) matrices permutable to ∗ matrices are characterized and counted. Commutivity of matrices which permute to ∗ matrices is also discussed