On the Structure of Graphs with Non-Surjective L(2,1)-Labelings

For a graph G, an L(2,1)-labeling of G with span k is a mapping L \right arrow \{0, 1, 2, \ldots, k\} such that adjacent vertices are assigned integers which differ by at least 2, vertices at distance two are assigned integers which differ by at least 1, and the image of L includes 0 and k. The minimum span over all L(2,1)-labelings of G is denoted $\lambda(G)$, and each L(2,1)-labeling with span $\lambda(G)$ is called a $\lambda$-labeling. For $h \in \{1, \ldots, k-1\}$, h is a hole of Lif and only if h is not in the image of L. The minimum number of holes over all $\lambda$-labelings is denoted $\rho(G)$, and the minimum k for which there exists a surjective L(2,1)-labeling onto {0,1, ..., k} is denoted $\mu(G)$. This paper extends the work of Fishburn and Roberts on $\rho$ and $\mu$ through the investigation of an equivalence relation on the set of $\lambda$-labelings with $\rho$ holes. In particular, we establish that $\rho \leq \Delta$. We analyze the structure of those graphs for which $\rho \in \{ \Delta-1, \Delta \}$, and we show that $\mu = \lambda+ 1$ whenever $\lambda$ is less than the order of the graph. Finally, we give constructions of connected graphs with $\rho = \Delta$ and order $t(\Delta + 1)$, $1 \leq t \leq \Delta$

Covering the Boundary of a Simple Polygon with Geodesic Unit Disks

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm

On Ramsey Theory and Slow Bootstrap Percolation

This dissertation concerns two sets of problems in extremal combinatorics. The major part, Chapters 1 to 4, is about Ramsey-type problems for cycles. The shorter second part, Chapter 5, is about a problem in bootstrap percolation. Next, we describe each topic more precisely. Given three graphs G, L1 and L2, we say that G arrows (L1, L2) and write G → (L1, L2), if for every edge-coloring of G by two colors, say 1 and 2, there exists a color i whose color class contains Li as a subgraph. The classical problem in Ramsey theory is the case where G, L1 and L2 are complete graphs; in this case the question is how large the order of G must be (in terms of the orders of L1 andL2) to guarantee that G → (L1, L2). Recently there has been much interest in the case where L1 and L2 are cycles and G is a graph whose minimum degree is large. In the past decade, numerous results have been proved about those problems. We will continue this work and prove two conjectures that have been left open. Our main weapon is Szemeredi\u27s Regularity Lemma.Our second topic is about a rather unusual aspect of the fast expanding theory of bootstrap percolation. Bootstrap percolation on a graph G with parameter r is a cellular automaton modeling the spread of an infection: starting with a set A0, cointained in V(G), of initially infected vertices, define a nested sequence of sets, A0 ⊆ A1 ⊆. . . ⊆ V(G), by the update rule that At+1, the set of vertices infected at time t + 1, is obtained from At by adding to it all vertices with at least r neighbors in At. The initial set A0 percolates if At = V(G) for some t. The minimal such t is the time it takes for A0 to percolate. We prove results about the maximum percolation time on the two-dimensional grid with parameter r = 2

On locally rainbow colourings

Given a graph $H$, let $g(n,H)$ denote the smallest $k$ for which the following holds. We can assign a $k$-colouring $f_v$ of the edge set of $K_n$ to each vertex $v$ in $K_n$ with the property that for any copy $T$ of $H$ in $K_n$, there is some $u\in V(T)$ such that every edge in $T$ has a different colour in $f_u$. The study of this function was initiated by Alon and Ben-Eliezer. They characterized the family of graphs $H$ for which $g(n,H)$ is bounded and asked whether it is true that for every other graph $g(n,H)$ is polynomial. We show that this is not the case and characterize the family of connected graphs $H$ for which $g(n,H)$ grows polynomially. Answering another question of theirs, we also prove that for every $\varepsilon>0$, there is some $r=r(\varepsilon)$ such that $g(n,K_r)\geq n^{1-\varepsilon}$ for all sufficiently large $n$. Finally, we show that the above problem is connected to the Erd\H{o}s-Gy\'arf\'as function in Ramsey Theory, and prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each fixed $r$ the complete $r$-uniform hypergraph $K_n^{(r)}$ can be edge-coloured using a subpolynomial number of colours in such a way that at least $r$ colours appear among any $r+1$ vertices.Comment: 12 page