Extremal Colorings and Independent Sets

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. It was conjectured that extremal graphs are those which have clique number k and size (k2)+n−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(k2)+n−k(k2)+n−k. We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is Kk∪En−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eKk∪En−kKk∪En−k, K1∨(Kk−1∪En−k) role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eK1∨(Kk−1∪En−k)K1∨(Kk−1∪En−k) and (K1∨(Kk−1∪En−k−c+1))∪Ec−1 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(K1∨(Kk−1∪En−k−c+1))∪Ec−1(K1∨(Kk−1∪En−k−c+1))∪Ec−1 respectively

Extremes of the internal energy of the Potts model on cubic graphs

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a conjecture of Galvin and Tetali

Extremal H-Colorings of Graphs with Fixed Minimum Degree

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph Kδ,n-δ is the n-vertex graph with minimum degree δ that has the largest number of independent sets. In this paper, we begin the project of generalizing this result to arbitrary H. Writing hom(G, H) for the number of H-colorings of G, we show that for fixed H and δ = 1 or δ = 2, hom(G, H) ≤ max{hom(Kδ+1,H)n⁄δ =1, hom(Kδ,δ,H)n⁄2δ, hom(Kδ,n-δ,H)} for any n-vertex G with minimum degree δ (for sufficiently large n). We also provide examples of H for which the maximum is achieved by hom(Kδ+1, H)n⁄δ+1 and other H for which the maximum is achieved by hom(Kδ,δ,H)n⁄2δ. For δ ≥ 3 (and sufficiently large n), we provide a infinite family of H for which hom(G, H) ≤ hom (Kδ,n-δ, H) for any n-vertex G with minimum degree δ. The results generalize to weighted H-colorings