On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents, point by point, a partially ordered set. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. There are many recent results about variants of the general on-line chain partition problem. With this survey we attempt to give an overview over

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One of the simplest heuristics for obtaining a proper coloring of a graph is the first-fit algorithm. First-fit visits each vertex of the graph in the specified order and assigns to every point the least possible number. Let G be a class of incomparability graphs with bounded maximum clique size, closed under taking induced subgraphs. We prove that first-fit uses a bounded number of colors on the graphs in G iff there is an incomparability graph of clique size 2 not contained in G.

An additive coloring of a graph G is an assignment of positive integers {1, 2, . . . , k} to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by η(G). We prove that η(G) 468 for every planar graph G. This improves a previous bound η(G) 5544 due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that η(G) 36 for 3-colorable planar graphs, and η(G) 4 for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each r 2 there is an r -chromatic graph G r with no additive coloring by elements of any abelian group of order r .

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. We analyze the chain partition game for up-growing semi-orders. Surprisingly, the golden ratio comes into play and the value of the game isw .