Extremal traceable graphs with non-traceable edges

Adam Paweł Wojda
2009 Opuscula Mathematica  
By NT(n) we denote the set of graphs of order n which are traceable but have non-traceable edges, i.e. edges which are not contained in any hamiltonian path. The class NT(n) has been considered by Balińska and co-authors in a paper published in 2003, where it was proved that the maximum size tmax(n) of a graph in NT(n) is at least (n 2 − 5n + 14)/2 (for n ≥ 12). The authors also found tmax(n) for 5 ≤ n ≤ 11. We prove that, for n ≥ 5, tmax(n) = max{'n −2 2´+ 4,'n − n−1 2 2´+ n−1 2 2 } and,
more » ... −1 2 2 } and, moreover, we characterize the extremal graphs (in fact we prove that these graphs are exactly those already described in the paper by Balińska et al.). We also prove that a traceable graph of order n ≥ 5 may have at most n−3 2 n−3 2 non traceable edges (this result was conjectured in the mentioned paper by Balińska and co-authors).
doi:10.7494/opmath.2009.29.1.89 fatcat:nerhbetakfcxtjfxgp7guebbgi