H-coloring problem is a coloring problem with restrictions such that some pairs of colors cannot be used for adjacent vertices, where H is a graph representing the restrictions of colors. We deal with the case that H is the complement graph of a cycle of odd order 2p + 1. This paper presents the following results: (1) chordal graphs and internally maximal planar graphs are -colorable if and only if they are p-colorable (p 2), (2) -coloring problem on planar graphs is NP-complete, and (3) there exists a class that includes infinitely many -colorable but non-3-colorable planar graphs.

Coloring problem is a well-known combinatorial optimization problem of graphs. This paper considers H-coloring problems, which are coloring problems with restrictions such that some pairs of colors can not be used for adjacent vertices. The restriction of adjacent colors can be represented by a graph H, i.e., each vertex represents a color and each edge means that the two colors corresponding to the two end-vertices can be used for adjacent vertices. Especially, H-coloring problem with a complete graph H of order k is equivalent to the traditional k-coloring problem. This paper presents sufficient conditions such that H-coloring problem can be reduced to an H-coloring problem, where H is a subgraph of H. And it shows a hierarchy about classes of H-colorable graphs for any complement graph H of a cycle of order odd n 5.