It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere-zero 4-flow. If both factors are bipartite, then the product admits a nowhere-zero 3-flow.
In the quest to better understand the connection between median graphs, triangle-free graphs and partial cubes, a hierarchy of subclasses of partial cubes has been introduced. In this article, we study the role of tiled partial cubes in this scheme. For instance, we prove that ------------------
It is well known that every planar graph G is 2-colorable in such a way that no 3-cycle of G is monochromatic. In this paper, we prove that G has a 2-coloring such that no cycle of length 3 or 4 is monochromatic. The complete graph K 5 does not admit such a coloring. On the other hand, we extend the result to K 5 -minor-free graphs. There are planar graphs with the property that each of their 2-colorings has a monochromatic cycle of length 3, 4, or 5. In this sense, our result is best possible. ß
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