Given a family of r-uniform hypergraphs F (or r-graphs for brevity), the Turán number ex(n, F ) of F is the maximum number of edges in an r-graph on n vertices that does not contain any member of F . A pair {u, v} is covered in a hypergraph G if some edge of G contains {u, v}.

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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 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].

An additive coloring of a graph G is a labeling of the vertices of G from {1, 2,. .. , k} such that two adjacent vertices have distinct sums of labels on their neighbors. The least integer k for which a graph G has an additive coloring is called the additive coloring number of G, denoted χ Σ (G). Additive coloring is also studied under the names lucky labeling and open distinguishing. In this paper, we improve the current bounds on the additive coloring number for particular classes of graphs by proving results for a list version of additive coloring. We apply the discharging method and the Combinatorial Nullstellensatz to show that every planar graph G with girth at least 5 has χ Σ (G) ≤ 19, and for girth at least 6, 7, and 26, χ Σ (G) is at most 9, 8, and 3, respectively. In 2009, Czerwiński, Grytczuk, andŻelazny conjectured that χ Σ (G) ≤ χ(G), where χ(G) is the chromatic number of G. Our result for

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