Coloring immersion-free graphs

A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. The notion of an immersion is quite similar to the well-known notion of a minor, as structural approach inspired by the theory of graph minors has been extremely successful in immersions. Hadwiger's conjecture on graph coloring, generalizing the Four Color Theorem, states that every loopless graph without a K_k-minor is (k−1)-colorable, where K_k is the complete graph on k vertices. This is a long standing open problem in graph theory, and it is even unknown whether it is possible to determine ck-colorability of K_k-minor-free graphs in polynomial time for some constant c. In this paper, we address coloring graphs without H-immersion. In contrast to coloring H-minor-free graphs, we show the following: 1. there exists a fixed-parameter algorithm to decide whether or not a given graph G without an immersion of a graph H of maximum degree d is (d−1)-colorable, where the size of H is a parameter. In fact, if G is (d−1)-colorable, the algorithm produces such a coloring, and2. for any positive integer k (k≥6), it is NP-complete to decide whether or not a given graph G without a K_k-immersion is (k−3)-colorable.