Maximizing H-Colorings of Connected Graphs with Fixed Minimum Degree

A Degree Condition For Graphs to Have Connected (g, f)-Factors

Some Problems Involving H-colorings of Graphs

by John Alan Engbers For graphs G and H, an H-coloring of G, or homomorphism from G to H, is an edge-preserving map from the vertices of G to the vertices of H. H-colorings generalize such graph theory notions as proper colorings and independent sets. In this dissertation, we consider four questions involving H-colorings of graphs. Recently, Galvin [27] showed that the maximum number of indepen...

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...

Extremal H-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 res...

The irregularity and total irregularity of Eulerian graphs

For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.