The square of a planar cubic graph is 77-colorable

We prove the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable. Here, $7$ cannot be replaced by $6$

On the number of Hamiltonian cycles in tournaments

AbstractThe main results assert that the minimum number of Hamiltonian bypasses in a strong tournament of order n and the minimum number of Hamiltonian cycles in a 2-connected tournament of order n increase exponentially with n. Furthermore, the number of Hamiltonian cycles in a tournament increases at least exponentially with the minimum outdegree of the tournament. Finally, for each k⩾1 there are infinitely many tournaments with precisely k Hamiltonian cycles

Hypohamiltonian and hypotraceable graphs

AbstractIn this note hypohamiltonian and hypotraceable graphs are constructed

Sign-nonsingular matrices and even cycles in directed graphs

AbstractA sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (aij) such that at least one of aij and aij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and aij=0 iff aji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects

Independent Dominating Sets and a Second Hamiltonian Cycle in Regular Graphs

AbstractIn 1975, John Sheehan conjectured that every Hamiltonian 4-regular graph has a second Hamiltonian cycle. Combined with earlier results this would imply that every Hamiltonianr-regular graph (r⩾3) has a second Hamiltonian cycle. We shall verify this forr⩾300