For a family of graphs , a graph is called -free if it does not contain any member of as a subgraph. The generalized Turán number
In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is Hamiltonian. This result implies that every more than -tough planar graph on at least three vertices is Hamiltonian and so has a 2-factor. Owens in 1999 constructed non-Hamiltonian maximal planar graphs of toughness arbitrarily close to and asked whether there exists a maximal non-Hamiltonian planar graph of toughness exactly . In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by cons
Answering a question of Erdős and Nešetřil, we show that the maximum number of inclusion-wise minimal vertex cuts in a graph on vertices is at most for large enough .
A graph is hypohamiltonian if it is non-hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay, Östergård, Pettersson, and Zamfirescu. That result is here improved upon by two planar hypohamiltonian graphs on 34 vertices. We exploited a special subgraph contained in two graphs of Jooyandeh et al., and modified it to construct the two 34-vertex graphs and six planar hypohamiltonian graphs on 37 vertices. Each of the 34-vertex graphs has 26 cubic vertices, improving upon the result of Jooyandeh et al. that planar hypohamiltonian graphs have 30 cubic vertices. We use the 34-vertex graphs to construct hypohamiltonian graphs of order 34 with crossing number 1, improving the best-known bound of 36 due to Wiener. Whether there exists a planar hypohamiltonian graph on 41 vertices was an open question. We settled this question by applying an operation introduced by Thomassen to the 37-vertex graphs to obtain several planar hypohamiltonian graphs on 41 vertices. The 25 planar hypohamiltonian graphs on 40 vertices of Jooyandeh et al. have no nontrivial automorphisms. The result is here improved upon by six planar hypohamiltonian graphs on 40 vertices with nontrivial automorphisms.
The square of a graph , denoted , has the same vertex set as and has an edge between two vertices if the distance between them in is at most 2. In general, for every graph . Charpentier (2014) asked whether
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