Extremal Graphs for Widom–Rowlinson Colorings in k-Chromatic Graphs

• Published in: Graphs and combinatorics Vol. 40; no. 6; p. 134
• Main Authors: Engbers, John, Erey, Aysel
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.12.2024; Springer Nature B.V
• Subjects: Apexes; Combinatorics; Connectivity; Engineering Design; Equality; Graph theory; Graphs; Homomorphisms; Hypotheses; Mathematics; Mathematics and Statistics; Original Paper; 05C60; Widom–Rowlinson coloring; 05C30; 05C40; Connectivity; Chromatic number; 05C15
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-024-02869-3

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 ).