Es wird ein zweifach zusammenhängender (nichtplanarer) Graph angegeben, der keine zwei Knotenpunkte besitzt, so daß jeder längste Kreis des Graphen durch weinigstens einen der beiden Knotenpunkte geht.
Es wird ein zweifach zusammenhängender (nichtplanarer) Graph angegeben, der keine zwei Knotenpunkte besitzt, so daß jeder längste Kreis des Graphen durch weinigstens einen der beiden Knotenpunkte geht.
Tucker's lemma is a combinatorial result which may be used to derive several theorems in topology. Some basic properties are established for the cube of integer lattice points. Tucker's lemma is then proved by applying a result which was originally presented for the octahedral subdivision of the n-disk.
Abstract
The four-color problem concerning planar graphs is shown to have meaningful higher-dimensional analogs.
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of whenever the latter is congruent to 1, 7 or 10 modulo 12.
The concept of coloring a graph has been shown to be subsumed by that of an homomorphism. This led in [3] to the definition of a complete n-coloring of a graph G and suggested therefore a new invariant, which we now call the “achromatic number” ψ(G). While the chromatic number χ(G) is the minimum number of colors required for (a complete coloring of) the points of G, the achromatic number is the maximum such number. We obtain several bounds for ψ(G) in terms of other invariants of a graph, and in particular we show that, for any graph G having p points, x(G)+ͨ(G)¯⩽p+1, a result which generalizes a theorem of Nordhaus and Gaddum [4].
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of whenever the latter is congruent to 4 modulo 12.
This paper gives a proof of the fact that the chromatic number of an orientable surface of genus p is equal to the integral part of whenever the latter is congruent to 3, 5, 6, or 9 modulo 12.
It has been shown by M. Marcus and others that, in regard to combinatorial matrix functions and combinatorial inequalities, it is frequently fruitful to pass immediately from the consideration of permutations to the consideration of their tensor representations. Such an approach embeds the combinatorial arguments into the framework of linear algebra and frequently results in deeper theorems. It is interesting to note that certain basic combinatorial identities concerned with pattern enumeration and combinatorial generating functions can also be put into this framework. In this paper we consider one possible way of doing this.
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