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We describe how one department moved from skepticism to wide-spread embrace of coordinated Calculus I. Data played a crucial role in this process. Details of this journey, including successful stra...

An edge-colored graph is properly connected if for every pair of vertices u and v there exists a properly colored uv-path (i.e. a uv-path in which no two consecutive edges have the same color). The proper connection number of a connected graph G, denoted pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is properly connected. An edge-colored graph is flexibly connected if for every pair of vertices u and v there exist two properly colored paths between them, say P and Q, such that the first edges of P and Q have different colors and the last edges of P and Q have different colors. The flexible connection number of a connected graph G, denoted fpc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is flexibly connected. In this paper, we demonstrate several methods for constructing graphs with pc(G) = 2 and fpc(G) = 2. We describe several families of graphs such that pc(G) ≥ 2 and we settle a conjecture from [BFG+12]. We prove that if G is connected and bipartite, then pc(G) = 2 is equivalent to being 2-edge-connected and fpc(G) = 2 is equivalent to the existence of a path through all cut-edges. Finally, it is proved that every connected, k-regular, Class 1 graph has flexible connection number 2.

We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and k-partite results where they exist. We then prove that if G is a balanced tripartite graph on 3n vertices, G must contain a cycle of length at least 3n − 1, provided that e(G) ≥ 3n2 − 4n + 5 and n ≥ 14.

An \emph{obstacle representation} of a graph $G$ is a straight-line drawing of $G$ in the plane together with a collection of connected subsets of the plane, called \emph{obstacles}, that block all non-edges of $G$ while not blocking any of the edges of $G$. The \emph{obstacle number} obs$(G)$ is the minimum number of obstacles required to represent $G$. We study the structure of graphs with obstacle number greater than one. We show that the icosahedron has obstacle number $2$, thus answering a question of Alpert, Koch, \& Laison asking whether all planar graphs have obstacle number at most $1$. We also show that the $1$-skeleton of a related polyhedron, the \emph{gyroelongated $4$-dipyramid}, has obstacle number $2$. The order of this graph is $10$, which is also the order of the smallest known graph with obstacle number $2$. Some of our methods involve instances of the Satisfiability problem, we make use of various "SAT solvers" in order to produce computer-assisted proofs., 19 pages, 12 figures. Revision: fixed the proof of Lemma 5.1; rearranged some sections slightly and added the 4-point rule; added a new figure showing the obstacle number of the dodecahedron is 1

Given a family of (hyper)graphs $\mathcal{F}$ a (hyper)graph $G$ is said to be $\mathcal{F}$-saturated if $G$ is $F$-free for any $F \in\mathcal{F}$ but for any edge e in the complement of $G$ the (hyper)graph $G + e$ contains some $F\in\mathcal{F}$. We survey the problem of determining the minimum size of an $\mathcal{F}$-saturated (hyper)graph and collect many open problems and conjectures.