Saturation numbers in tripartite graphs

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the minimum number of edges in an $F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\ell\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-2})$ within an additive constant. We also include general constructions of $K_{\ell,m,p}$-saturated subgraphs of $K_{n_1,n_2,n_3}$ with few edges for $\ell\ge m\ge p>0$.Comment: 18 pages, 6 figure

Planar digraphs without large acyclic sets

Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.Comment: 3 pages, 1 figur

Three Existence Problems in Extremal Graph Theory

Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics. In this thesis we take this approach to three different structural questions rooted in extremal graph theory. When studying graph representations, we seek efficient ways to encode the structure of a graph. For example, an {\it interval representation} of a graph $G$ is an assignment of intervals on the real line to the vertices of $G$ such that two vertices are adjacent if and only if their intervals intersect. We consider graphs that have {\it bar $k$-visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations. We obtain results on $\mathcal{F}_k$, the family of graphs having bar $k$-visibility representations. We also study $\bigcup_{k=0}^{\infty} \mathcal{F}_k$. In particular, we determine the largest complete graph having a bar $k$-visibility representation, and we show that there are graphs that do not have bar $k$-visibility representations for any $k$. Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs. Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around $G$ provided that each move transfers weight from a vertex to a heavier neighbor. Our goal in making acquisition moves is to consolidate all of the weight in $G$ on the minimum number of vertices; this minimum number is the {\it acquisition number} of $G$. We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight. We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter $2$. We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex. Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects. Some local conditions are so limiting that very few objects satisfy the requirements. For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor. Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex. We study a related structural restriction where similar phenomena occur. For a fixed graph $H$, we consider those graphs that do not contain $H$ and such that the addition of any edge completes exactly one copy of $H$. Such a graph is called {\it uniquely $H$-saturated}. We study the existence of uniquely $H$-saturated graphs when $H$ is a path or a cycle. In particular, we determine all of the uniquely $C_4$-saturated graphs; there are exactly ten. Interestingly, the uniquely $C_{5}$-saturated graphs are precisely the friendship graphs characterized by Wilf

Graph Saturation in Multipartite Graphs

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal F}$ is a subgraph of $J+e$. We let $\text{ex}({\mathcal F},G)$ and $\text{sat}({\mathcal F},G)$ denote the maximum and minimum size of an ${\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\mathcal F}$ is a subgraph of $G$, then $\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|$. In this paper, for $k\ge 3$ and $n\ge 100$ we determine $\text{sat}(K_3,K_k^n)$, where $K_k^n$ is the complete balanced $k$-partite graph with partite sets of size $n$. We also give several families of constructions of $K_t$-saturated subgraphs of $K_k^n$ for $t\ge 4$. Our results and constructions provide an informative contrast to recent results on the edge-density version of $\text{ex}(K_t,K_k^n)$ from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure

List Distinguishing Parameters of Trees

A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, we say that G is (properly) L-distinguishable if there is a (proper) distinguishing coloring f of G such that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is the minimum integer k such that G is L-distinguishable for any list assignment L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is properly L-distinguishable for any list assignment L with |L(v)| = k for all v. In this paper, we study these distinguishing parameters for trees, and in particular extend an enumerative technique of Cheng to show that for any tree T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page