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; k₅ 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 k₅,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); K₄ is such a graph.