On the Fiedler value of large planar graphs

The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $\lambda_{2\max}$, and we show the bounds $2+\Theta(\frac{1}{n^2}) \leq \lambda_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $\lambda_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and Its Application

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

Planarity is (almost) locally checkable in constant-time

Locally checkable proofs for graph properties were introduced by G\"o\"os and Suomela \cite{GS}. Roughly speaking, a graph property \cP is locally checkable in constant-time, if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are indeed vertices of a graph having the given property. For a given \eps>0, we call a property \cP \eps-locally checkable in constant-time if the vertices of a graph having the given property can be convinced at least that they are in a graph \eps-close to the given property. We say that a property \cP is almost locally checkable in constant-time, if for all \eps>0, \cP is \eps-locally checkable in constant-time. It is not hard to see that in the universe of bounded degree graphs planarity is not locally checkable in constant-time. However, the main result of this paper is that planarity of bounded degree graphs is almost locally checkable in constant-time. The proof is based on the surprising fact that although graphs cannot be convinced by their planarity or hyperfiniteness, planar graphs can be convinced by their own hyperfiniteness. The reason behind this fact is that the class of planar graphs are not only hyperfinite but possesses Property A of Yu.Comment: 13 page