Ramsey theory
Connectivity, graph minors, and subgraph multiplicity.
D. Eppstein.
Tech. Rep. 92-06, ICS, UCI, 1992.
J. Graph Th. 17: 409–416, 1993.It was known that planar graphs have \(O(n)\) subgraphs isomorphic to \(K_3\) or \(K_4\). That is, \(K_3\) and \(K_4\) have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.
More generally, let \(\mathcal{F}\) be a minor-closed family, and let \(x\) be the smallest number such that some complete bipartite graph \(K_{x,y}\) is a forbidden minor for \(\mathcal{F}\). Then the \(x\)-connected graphs have linear subgraph multiplicity for \(\mathcal{F}\), and there exists an \((x-1)\)-connected graph (namely \(K_{x-1,x-1}\) that does not have linear subgraph multiplicity. When \(x\le 3\) or when \(x=4\) and the minimal forbidden minors for \(\mathcal{F}\) are triangle-free, then the graphs with linear subgraph multiplicity for \(\mathcal{F}\) are exactly the \(x\)-connected graphs.
Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.
New algorithms for minimum area k-gons.
D. Eppstein.
Tech. Rep. 91-59, ICS, UCI, 1991.
3rd ACM-SIAM Symp. Discrete Algorithms, Orlando, 1992, pp. 83–86.Shows that the minimum area polygon containing k of n points must be near a line determined by two points, and uses this observation to find the polygon quickly. Merged with "Iterated nearest neighbors and finding minimal polytopes" in the journal version.
Separating geometric thickness from book thickness.
D. Eppstein.
arXiv:math.CO/0109195.We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness > t.
Separating thickness from geometric thickness.
D. Eppstein.
arXiv:math.CO/0204252.
10th Int. Symp. Graph Drawing, Irvine, 2002.
Springer, Lecture Notes in Comp. Sci. 2528, 2002, pp. 150–161.
In Towards a Theory of Geometric Graphs, AMS, 2004, Contemporary Math 342, J. Pach, ed., pp. 75–86.We show that thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness > t. The proof uses Ramsey-theoretic arguments similar to those in "Separating book thickness from thickness".
Stack-number is not bounded by queue-number.
V. Dujmović, D. Eppstein, R. Robert Hickingbotham, P. Morin, and D. R. Wood.
arXiv:2011.04195.
Combinatorica 42: 151–164, 2022.Stack number is also known as page number or book thickness; it is the minimum number of stacks needed so that you can process the vertices of a graph in some sequence, pushing each edge onto one of the stacks when you process its first endpoint and popping it from the same stack when you process its second endpoint. Queue number is defined in the same way using queues instead of stacks. We show that the strong products of triangular grids and high-degree stars have bounded queue number but unbounded stack number. This result disproves the Blankenship–Oporowski conjecture, according to which subdividing edges of a graph a constant number of times cannot decrease its stack number from non-constant to constant, because subdivisions of the same products also have bounded stack number. It also confirms a conjecture of Bonnet et al on the existence of graphs with bounded sparse twin-width and unbounded stack number.