For a fixed graph H, in the graph homomorphism problem, denoted by Hom(H), we are given a graph G and we have to determine whether there exists an edge-preserving mapping φ:V(G)→V(H)
Nearby in the stack
. Note that
Hom(C3)
, where
C3
is the cycle of length
3
, is equivalent to
3
-Coloring. The question whether
3
-Coloring is polynomial-time solvable on diameter-
2
graphs is a well-known open problem. In this paper we study the
Hom(C2k+1)
problem on bounded-diameter graphs for
k≥2
, so we consider all other odd cycles than
C3
. We prove that for
k≥2
, the
Hom(C2k+1)
problem is polynomial-time solvable on diameter-
(k+1)
graphs -- note that such a result for
k=1
would be precisely a polynomial-time algorithm for
3
-Coloring of diameter-
2
graphs. Furthermore, we give subexponential-time algorithms for diameter-
(k+2)
graphs. We complement these results with a lower bound for diameter-
(2k+2)
graphs -- in this class of graphs the
Hom(C2k+1)
problem is NP-hard and cannot be solved in subexponential-time, unless the ETH fails. Finally, we consider another direction of generalizing
3
-Coloring on diameter-
2
graphs. We consider other target graphs
H
than odd cycles but we restrict ourselves to diameter
2
. We show that if
H
is triangle-free, then
Hom(H)
is polynomial-time solvable on diameter-
2
graphs.
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