Selected topics in fractional graph theory

We may formulate the chromatic number {dollar}chi(G){dollar} of a graph G as follows: choose as few independent sets as possible such that every vertex of G is in at least one of the chosen sets. This is easily written as a {dollar}{lcub}0,1{rcub}{dollar}-integer program. We define the fractional chromatic number {dollar}chisb{lcub}f{rcub}(G){dollar} to be the value of the linear relaxation of this program. The integer dual of this program yields the clique number {dollar}omega(G){dollar} of the graph, and we define fractional clique number {dollar}omegasb{lcub}f{rcub}(G){dollar} to be the value of the linear relaxation of the integer dual. By strong linear programming duality, {dollar}omega(G)leqomegasb{lcub}f{rcub}(G)=chisb{lcub}f{rcub}(G)leqchi(G){dollar} for any finite graph.; We may equivalently write {dollar}chisb{lcub}f{rcub}(G)={lcub}rm lim{rcub}sb{lcub}btoinfty{rcub}(G)/b,{dollar} where {dollar}chisb{lcub}b{rcub}(G){dollar} is the fewest total colors need in order to assign each vertex a set of b distinct colors disjoint from the color sets of its neighbors.; In Chapter 2, for infinite graph G, we define {dollar}overline{lcub}chisb{lcub}f{rcub}{rcub}(G){dollar} to be the supremum of {dollar}chisb{lcub}f{rcub}{dollar} of all G's finite subgraphs. We answer an open problem of Leader (8) by constructing infinite graphs for which {dollar}overline{lcub}chisb{lcub}f{rcub}{rcub}