A note on uniquely H-colorable graphs

Abstract

For a graph H, we compare two notions of uniquely H-colourable graphs, where one is defined via automorphisms, the second by vertex partitions. We prove that the two notions of uniquely H-colourable are not identical for all H, and we give a condition for when they are identical. The condition is related to the first homomorphism theorem from algebra.1991 Mathematics Subject Classification. 05C15, 05C75.

The good, the bad, and the great: Homomorphisms and cores of random graphs

“…The graph P was the only such example given in [1]. We demonstrate that there are infinitely many good but not great cores in this section.…”

“…There are two natural definitions of a uniquely H-colourable graph. Following [1,6,7], a graph G is uniquely H-colourable if G is H-colourable so that every homomorphism from G to H is onto, and for all homomorphisms f , h from G to H, there is an automorphism g of H so that f = gh. On the other hand, a graph G is weakly uniquely H-colourable if a similar definition holds, but with g only required to be a bijection from V(H) to itself.…”

“…For many familiar cores H such as cliques, odd cycles, odd wheels, and the Petersen graph, the two notions of uniquely H-colourable coincide. However, as discussed in [1], there are infinitely many examples of graphs H where the class of weakly uniquely H-colourable graphs strictly contains the class of uniquely H-colourable graphs. Following the notation in [1], a core H is good if the two notions uniquely colourable coincide; H is great if for all e ∈ E(H), there is some homomorphism from H − e to H that is not onto (or equivalently, not injective).…”

“…In [1], it was proved that every great core is good, but the converse fails for the Petersen graph. Great cores have the following algebraic characterization related to the first homomorphism theorem.…”