A proper vertex coloring of a graphG is called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) ∈ L(v). If G is L-star-colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is called k-star-choosable. The star list chromatic number of G, ...

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses...

An axis-parallel d–dimensional box is a Cartesian product R1 × R2 × · · · × Rd where Ri (for 1 ≤ i ≤ d) is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension d, such that G is representable as the intersection graph of (axis–parallel) boxes in d–dimensional space. The concept of boxicity finds applications in various areas such a...

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree∆ are list 2-distance (∆ + 2)-colorable when ∆ ≥ 24 (Borodin and Ivanova (2...

In 1994, Erd?s, Gyárfás and ?uczak posed the following problem: given disjoint vertex sets V1,…,Vn of size k, with exactly one edge between any pair Vi,Vj, how large can n be such that there will always an independent transversal? They showed maximal is at most (1+o(1))k2, by providing explicit construction these parameters no transversal. also proved a lower bound which smaller 2e-factor. this...

Motivated by a longstanding conjecture of Thomassen, we study how large the average degree graph needs to be imply that it contains C4-free subgraph with at least t. Kühn and Osthus showed an bound which is double exponential in t sufficient. We give short proof this bound, before reducing single exponential. That is, show any G $$2^{ct^{2}\log\ t}$$ (for some constant c > 0) Finally, construct...