The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.We use an old topological result of Ky Fan [17] which generalizes the Borsuk-Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K t/2 , t/2 in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of t/2 + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur.

Shannon OR-capacity C OR (G) of a graph G, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter satisfying C OR (F ×G) ≤ min{C OR (F ), C OR (G)} for every pair of graphs, where F × G is the categorical product of graphs F and G. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on C OR (F × G) and elaborate on the question of how to find graph pairs for which it is known to be strictly less, than the upper bound min{C OR (F ), C OR (G)}. We present such graph pairs using the properties of Paley graphs.

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