An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3, 4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide sufficient conditions for the existence of such a subgraph.

Inspired by a 1987 result of Hanson and Toft [Edge‐colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge‐colored graphs. An edge‐coloring of a graph F is rainbow if every edge of F receives a different color. Let R(F) denote the set of rainbow‐colored copies of F. A t‐edge‐colored graph G is (R(F),t)‐saturated if G does not contain a rainbow copy of F but for any edge e∈E(G¯) and any color i∈[t], the addition of e to G in color i creates a rainbow copy of F. Let sat tfalse(n,frakturR(F)false) denote the minimum number of edges in an (R(F),t)‐saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph Kn lies between nlogn/loglogn and nlogn, the rainbow saturation number of an n‐vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.

Abstract:The well-known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a "universal friend"). V. Sós defined an analogous friendship property for 3-uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3-uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n = 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming.

Math 1190 Exam 1 Part 1 Retake Directions:• You have 25 minutes to complete this exam. The exam is worth 35 points (out of 100) of your Exam 1 grade.

A p-page embedding of a graph G is a vertex-ordering π of V (G) (along the "spine" of a book) and an assignment of edges to p half-planes (called "pages") such that no page contains crossing edges (alternating endpoints) relative to π. The pagenumber of G is the least p such that G has a p-page embedding. We disprove a conjecture of Ganley and Heath by showing that when k ≥ 3, there are k-trees that do not embed in k pages. We also present an algorithm that produces k-page embeddings for k-trees in a special class.