1,096 research outputs found
The square of a planar cubic graph is -colorable
We prove the conjecture made by G.Wegner in 1977 that the square of every
planar, cubic graph is -colorable. Here, cannot be replaced by
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 kâ©ľ1 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 mĂ— n matrix such that the columns of any real mĂ—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 nĂ—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
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