- September 10 -- Organizational meeting

Eric Andrews: The Division Algorithm --- Graph Theory Style

##### Abstract: A graph theoretic version of the division algorithm is described and some results are presented.

- September 17 --
Heather Jordon: Cycle Decompositions of Complete Graphs and Circulants

##### Abstract: In this talk, we will discuss several different methods for decomposing complete graphs or almost complete graphs into cycles of a fixed length. We will also discuss how these techniques apply to circulant graphs.

- September 24 -- Two 20-minute talks

Daniel Johnston: Twin Edge Colorings

##### Abstract: A twin edge k-coloring of a graph G is an edge coloring of G using as its colors the elements of the set of integers modulo k and possessing a certain prescribed double requirement. Results and questions concerning twin edge colorings are presented.

Chira Lumduanhom: On Monochromatic Colorings

##### Abstract: For a nontrivial connected graph G, vertex colorings of G are investigated that uses as its colors the elements of the set of integers modulo k and that generates a monochromatic vertex coloring of G (that assigns the same color to all vertices of G). This concept is inspired by the well-known Lights Out Puzzle and domination in graphs. Results and questions dealing with monochromatic colorings are presented.

- October 1 -- Allen Schwenk:
Independence Sequences in a Graph, Constrained and Unconstrained
##### Abstract: For a given graph G, we consider two sequences. In the vertex independence sequence a_1, a_2, ..., a_m each a_i denotes the number of ways to select a set of $i$ independent vertices in G, and m is the maximum order of an independent set of vertices. The edge independence sequence b_1, b_2, ..., b_m is defined similarly, and here m is the maximum size of an independent set of edges. It has been known for some time that the edge sequence must be unimodal, whereas the vertex sequence is totally unconstrained. We examine the edge sequence to find additional constraints beyond the required unimodality.

- November 5 -- Andrzej Dudek On generalized Ramsey numbers of Erdos and Rogers
##### Abstract: In a graph $G$, a set $S \subseteq V(G)$ is called $s$\emph{-independent} if the subgraph of $G$ induced by $S$ does not contain $K_s$. Let the $s$\emph{-independence number} of $G$, denoted by $\alpha_s(G)$, be the size of the largest $s$-independent set in $G$. (Hence, in particular, $\alpha_2(G)$ is the ordinary independence number.) The classical \emph{Ramsey number} $R(t,u)$ can be defined in this language as the smallest integer~$n$ such that $\alpha_2(G) \geq u$ for every $K_t$-free graph $G$ of order~$n$. A more general problem results by replacing the ordinary independence number by the $s$-independence number. Following this approach, in 1962 Erd{\H o}s and Rogers introduced the function $$ f_{s,t}(n)=\min \{\alpha_s(G): G \text{ is a $K_t$-free graph of order $n$}\}. $$ In this talk, we present some old and recent developments concerning this function. In particular, we partially confirm an old conjecture of Erd\H{o}s by showing that $\lim_{n\to \infty} \frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)}=\infty$ for any~$s\ge 4$ (joint work with John Retter and Vojta R\"odl). Furthermore, we discuss some extensions for hypergraphs (joint work with Dhruv Mubayi).

- November 12 -- Shelley Speiss On the Hamiltonicity of Random Graphs and Hypergraphs
##### Abstract: Let $G_{n,p}$ be the binomial random graph of order~$n$, in which every possible edge occurs independently with probability~$p$. The problem of determining the threshold of the existence of Hamilton cycles (\textit{i.e.} cycles which visit each vertex) in the random graph $G_{n,p}$, originally raised by Erd\H{o}s and R\'enyi, was first addressed by Koml\'{o}s and Szemer\'{e}di. Their result was successively improved by P\'osa and later studied by several other researchers, including Ajtai, Bollob\'as, Koml\'{o}s, Korshunov, \L{}uczak, and Szemer\'{e}di. One of the most celebrated results in the theory of random graphs states that if $p=(\log n + \log\log n + \omega(n))/n$, where $\omega(n)$ is an arbitrarily slowly growing function, then $G_{n,p}$ is Hamiltonian with high probability. In this talk, we give a gentle introduction to this topic and in addition discuss some recent developments for hypergraphs.

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