H-colorings of large degree graphs

We consider the H-coloring problem on graphs with vertices of large degree. We prove that for H an odd cycle, the problem belongs to P. We also study the phase transition of the problem, for an infinite family of graphs of a given chromatic number, i.e. the threshold density value for which the problem changes from P to NP-complete. We extend the result for the case that the input graph has a logarithmic size of small degree vertices.As a corollary, we get a new result on the chromatic number; a new family of graphs, for which computing the chromatic number can be done in polynomial time.Postprint (published version

Extremal \u3cem\u3eH\u3c/em\u3e-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 Κ is the n-vertex graph with minimum degree δ that has the largest number of independent sets. In this article, 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(Kσ+1, H)^(n/2σ), hom(Kσ,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 and other H for which the maximum is achieved by hom(Kσ+1, H)^(n/2σ). For δ≥3 (and sufficiently large n), we provide an infinite family of H for which hom(G,H)≤ hom(Kσ,n-δ, H) for any n-vertex G with minimum degree δ. The results generalize to weighted H-colorings

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

problems in graph theory and probability

This dissertation is a study of some properties of graphs, based on four journal papers (published, submitted, or in preparation). In the first part, a random graph model associated to scale-free networks is studied. In particular, preferential attachment schemes where the selection mechanism is time-dependent are considered, and an infinite dimensional large deviations bound for the sample path evolution of the empirical degree distribution is found. In the latter part of this dissertation, (edge) colorings of graphs in Ramsey and anti-Ramsey theories are studied. For two graphs, G, and H, an edge-coloring of a complete graph is (G;H)-good if there is no monochromatic subgraph isomorphic to G and no rainbow (totally muticolored) subgraph isomorphic to H in this coloring. Some properties of the set of number of colors used by some (G;H)-colorings are discussed. Then the maximum element in this set when H is a cycle is studied

Extremal \u3cem\u3eH\u3c/em\u3e-Colorings of Trees and 2-connected Graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H -colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n -vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs