On colorability of graphs with forbidden minors along paths and circuits

Graphs distinguished by K(r)-minor prohibition limited to subgraphs induced by circuits have chromatic number bounded by a function f(r); precise bounds on f (r) are unknown. If minor prohibition is limited to subgraphs induced by simple paths instead of circuits, then for certain forbidden configurations, we reach tight estimates. A graph whose simple paths induce K(3,3)-minor free graphs is proven to be 6-colorable; K(5) is such a graph. Consequently, a graph whose simple paths induce planar graphs is 6-colorable. We suspect the latter to be 5-colorable and we are not aware of such 5-chromatic graphs. Alternatively, (and with more accuracy) a graph whose simple paths induce {K(5), K(3,3)(-)}-minor free graphs is proven to be 4-colorable (where K(3,3)(-) is the graph obtained from K(3,3) by removing a single edge); K(4) is such a graph. (C) 2011 Elsevier B.V. All rights reserved.