Extremal Graphs for Widom–Rowlinson Colorings in k-Chromatic Graphs - Engbers, John; Erey, Aysel (Graphs and combinatorics, 2024)
The Widom–Rowlinson graph, H WR , is the fully looped path on three vertices. Let hom ( G , H WR ) be the number of graph homomorphisms from G to H WR or, equivalently, the number of H WR -colorings of G . We investigate extremal graphs for hom ( G , H WR ) for G in the family of k -chromatic graphs subject to various connectivity requirements. In particular, we determine the graphs G maximizing hom ( G , H WR ) in the families of n -vertex k -chromatic graphs, n -vertex connected k -chromatic graphs, n -vertex k -chromatic graphs with c components, n -vertex k -chromatic graphs without isolated vertices (for all n , k , c ), and n -vertex k -chromatic ℓ -connected, or minimum degree ℓ , graphs (for all k and ℓ , when n is large enough compared to k and ℓ ). Lastly, we determine the graphs G minimizing hom ( G , H WR ) in n -vertex k -chromatic graphs and n -vertex graphs with connectivity ℓ (for all n , k , ℓ ), and in n -vertex 2-chromatic graphs with connectivity ℓ (for all ℓ , when n is large enough compared to ℓ ).