12 research outputs found
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
Complexity Results for the Empire Problem in Collection of Stars
International audienceIn this paper, we study the Empire Problem, a generalization of the coloring problem to maps on two-dimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of deciding if s colors are sufficient to color the vertices of the graph such that vertices of the same block have the same color and vertices of two adjacent blocks have different colors. In this paper, we prove that given a 5-regular graph, deciding if there exists a 4-coloration is NP-complete. Also, we propose conditional NP-completeness results for the Empire Problem when the graph is a collection of stars. A star is a graph isomorphic to K 1,q for some q ≥ 1. More exactly, we prove that for r ≥ 2, if the (2r − 1)-coloring problem in 2r-regular connected graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = 2r − 1 is NP-complete for forests of K 1, r . Moreover, we prove that this result holds for r = 2. Also for r ≥ 3, if the r-coloring problem in (r + 1)-regular graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = r is NP-complete for forests of K 1, 1 = K 2, i.e., forest of edges. Additionally, we prove that this result is valid for r = 2 and r = 3. Finally, we prove that these results are the best possible, that is for smallest value of s or r, the Empire Problem in these classes of graphs becomes polynomial
Martingales on trees and the empire chromatic number of random trees
Abstract. We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps whose dual planar graph is a tree, with empires formed by exactly r countries. We prove that, for each fixed r> 1, with probability approaching one as the size of the graph grows to infinity, the minimum number of colours for which a feasible solution exists takes one of seven possible values.
Topology of random 2-complexes
We study the Linial–Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p≪n −1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π 1(Y) is free and H 2(Y)=0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p≫n −1/2+ϵ , where ϵ>0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n→∞. We also establish several related results; for example, we show that for p<c/n with c<3 the fundamental group of a random 2-complex contains a non-abelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embeddings and immersions of 2-complexes into a random 2-complex
Three-Dimensional Grid Drawings with Sub-Quadratic Volume
A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n^{3/2}) volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n^2). These results (partially) solve open problems due to Felsner, Wismath, and Liotta [Graph Drawing 2001] and Pach, Thiele, and Toth [Graph Drawing 1997]