| This thesis studies DP-coloring of planar graphs,fractional DP-coloring of seriesparallel graphs and k-common list coloring version of Hadwiger conjecture.DP-coloring is a generalization of list coloring.For any graph G,the choice number ch(G)of G is bounded from above by the DP-chromatic number χDP(G).An(a,b)-bipartite graph is a bipartite graph with bipartition(X,Y)such that dG(x)≤a for every x ∈ X and dG(y)≤b for every y ∈ Y.This thesis proves that for Δ≥ 3,if F is the square of the line graph of a(2,Δ)-bipartite graph,then χDP(F)≤3 Δ-2.The factional DP-chromatic number of G is defined as χDP*(G)=inf{a/b:G is(a,b)-DP-colourable}.For any integer t,let Qt={G:G is a series-parallel graph with girth at least t}.This thesis proves that if t ∈ {4q-1,4q,4q+1,4q+2},thenχDP*(Qt)=2+1/q.Hadwiger conjecture asserts that every Kt-minor-free graph is(t-1)-colorable.The list version of Hadwiger conjecture is not ture.Voigt proved that there is a k5minor free graph which is not 4-choosable.Janos Baeat,Gwenael Joret and David R.Wood further proved that for every positive integer t,there is a K3t+2-minor-free graph that is not 4t-choosable.The k-common list chromatic number of a graph G,denoted by chk(G),is defined as the minimum positive integer t such that for every t-list assignment L of G,if |∩x∈V(G)L(v)|≥ k,then G is L-colourable.This thesis proves that for any positive integers k,t,if 4t≥ 2k-1,then there is a K3t+2-minor-free graph that is not k-common(4t-k+1)-choosable. |
