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List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t \geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| \geq k$ for all vertices $v$ and $|L(u)\cup L(v)| \geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable., Comment: 10 pages, 2 figures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at most $s$ common colors in their lists. Every planar graph is $(4,1)$-choosable, but there exist planar graphs that are not $(4,3)$-choosable. It is an open question whether planar graphs are always $(4,2)$-choosable. A chorded $\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for each $\ell \in \{5,6,7\}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $\ell$-cycles., Comment: 33 pages, 14 figures. Collaboration began in the Iowa State University Discrete Mathematics Working Seminar 2014-2015

A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219]., Comment: 11 pages, 6 figures

Given a graph $G$ and a positive integer $R$ we address the following combinatorial search theoretic problem: What is the minimum number of queries of the form "does an unknown vertex $v \in V(G)$ belong to the ball of radius $r$ around $u$?" with $u \in V(G)$ and $r\le R$ that is needed to determine $v$. We consider both the adaptive case when the $j$th query might depend on the answers to the previous queries and the non-adaptive case when all queries must be made at once. We obtain bounds on the minimum number of queries for hypercubes, the Erd\H os-R\'enyi random graphs and graphs of bounded maximum degree .

A $(v,k,t)$ packing of size $b$ is a system of $b$ subsets (blocks) of a $v$-element underlying set such that each block has $k$ elements and every $t$-set is contained in at most one block. $P(v,k,t)$ stands for the maximum possible $b$. A packing is called abundant if $b> v$. We give new estimates for $P(v,k,t)$ around the critical range, slightly improving the Johnson bound and asymptotically determine the minimum $v=v_0(k,t)$ when abundant packings exist. For a graph $G$ and a positive integer $c$, let $\chi_\ell(G,c)$ be the minimum value of $k$ such that one can properly color the vertices of $G$ from any assignment of lists $L(v)$ such that $|L(v)|=k$ for all $v\in V(G)$ and $|L(u)\cap L(v)|\leq c$ for all $uv\in E(G)$. Kratochv\'{\i}l, Tuza and Voigt in 1998 asked to determine $\lim_{n\rightarrow \infty} \chi_\ell(K_n,c)/\sqrt{cn}$ (if exists). Using our bound on $v_0(k,t)$, we prove that the limit exists and equals $1$. Given $c$, we find the exact value of $\chi_\ell(K_n,c)$ for infinitely many $n$., Comment: 6 pages

For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochv\'{i}l, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of \chi_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs., Comment: 9 pages