Dan Archdeacon

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It has been shown that every quadrangulation on any nonspherical orientable closed surface with a suf®ciently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface Nk has chromatic number at least 4 if G has a cycle of odd length which cuts open Nk into an orientable surface.(More)
'The 4-Color Theorem (4CT) states that for any connected, bridgeless graph embedded in the plane, one can properly 4-color the faces, i.e., assign one of 4 colors to each face (region) such that two faces which meet along a common edge receive distinct colors. Closely related to the 4CT is the concept of Tait colorings of a cubic graph, i.e., a 3coloring of(More)
A graph is Y Y-reducible if it can be reduced to a vertex by a sequence of series-parallel reductions and Y Y-transformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that four-terminal planar graphs are Y Y-reducible when at least three of the vertices lie on the same face. Using(More)
In this article we generalize both ordinary and permutation voltage constructions to obtain all branched coverings of maps. We approach a map as a set of flags together with three fixed-point-free involutions and relate this approach with other standard representations. We define a lift as a function from these flags into a group. Ordinary voltage and(More)
A cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is(More)