35 research outputs found
The number of labeled k-dimensional trees
AbstractThe graphs known as trees have natural analogues in higher dimensional simplicial complexes. As an extension of Cayley's formula nn−2, the number of these k-dimensional trees on n-labeled vertices is shown to be (nk) (kn−k2+1)n−k−2
On nonreconstructable tournaments
AbstractPairs of non-isomorphic strong tournaments of orders 5 and 6 are given for which the subtournaments of orders 4 and 5, respectively, are pairwise isomorphic. Herefore, only pairs of orders 3 and 4 were known
The Complexity of the Empire Colouring Problem
We investigate the computational complexity of the empire colouring problem
(as defined by Percy Heawood in 1890) for maps containing empires formed by
exactly countries each. We prove that the problem can be solved in
polynomial time using colours on maps whose underlying adjacency graph has
no induced subgraph of average degree larger than . However, if , the problem is NP-hard even if the graph is a forest of paths of arbitrary
lengths (for any , provided .
Furthermore we obtain a complete characterization of the problem's complexity
for the case when the input graph is a tree, whereas our result for arbitrary
planar graphs fall just short of a similar dichotomy. Specifically, we prove
that the empire colouring problem is NP-hard for trees, for any , if
(and polynomial time solvable otherwise). For arbitrary
planar graphs we prove NP-hardness if for , and , for . The result for planar graphs also proves the NP-hardness of colouring
with less than 7 colours graphs of thickness two and less than colours
graphs of thickness .Comment: 23 pages, 12 figure
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
On the honesty of graph complements
AbstractA graph is called honest if its edge-integrity equals its order. It is shown in this paper that except for the path of length 3, every graph that is not honest has an honest complement. This result is extended to complements of products and applied to the Nordhaus-Gaddum theory for edge- integrity
Bounds and Methods for k-Planar Crossing Numbers
The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K_{2k+1,q}, for k >= 2. We prove tight bounds for complete graphs