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Abstract:A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color.

Let F be an r-uniform hypergraph and G be a multigraph. The hypergraph subhypergraph that is isomorphic to a Berge-G. We prove bounds on the maximum number of edges in an r-uniform linear hypergraph that is K 2,t -free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is {C 3 , K 2,3 }-free.

The classical Kővári-Sós-Turán theorem states that if G is an n-vertex graph with no copy of K s,t as a subgraph, then the number of edges in G is at most O(n 2−1/s ). We prove that if one forbids K s,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a nontrivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a nontrivial upper bound on the number of cliques of fixed order in a K r -free graph with no induced copy of K s,t . This result is an induced analog of a recent theorem of Alon and Shikhelman and is of independent interest. MSC: 05C35, 05C69

a b s t r a c tThe Turán number of a graph H, denoted ex(n, H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi (2011) conjectured that if H is any graph and ex(n, H) = O(n α ) where α > 1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(n α ) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemerédi (1978). Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.