On exhaustive reducible partition of graphs and its application to Hadwiger conjecture

An undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges. If G does not have a graph H as a minor, then we say that G is H-free. Hadwiger conjecture claim that the chromatic number of G may be closely related to whether it contains Kn+1 minors. To study the coloring of a Kn+1-free G, we propose a new concept of reducible partition of vertex set VG of G. A reducible partition(RP) of a graph G with Kn minors and without Kn+1 minors is defined as a two-tuples {S1 ⊆ VG, S2 ⊆ VG} which satisfy the following condisions: (1) S1 ∪ S2 = VG, S1 ∩ S2 = ∅ (2) S2 is dominated by S1, (3) the induced subgraph G [S1] is a forest, (4) the induced subgraph G [S2] is Kn-free. We will show that the reducible partition always exist and further we can obtain an exhaustive reducible partition(ERP) of VG: {S1, S2, · · · , Sm.m ≤ n−1} such that: (1) m ⋃ i=1 Si = VG, Si ∩ Sj = ∅ for i 6= j, (2) Sj is dominated by Si if i ≤ j (3) each induced subgraph G [Si] is a forest, (4) the induced subgraph G [ m ⋃ j=k Sj ] is Kn−k+2-free. ?Fully documented templates are available in the elsarticle package on CTAN. 1Since 1880. Preprint submitted to Journal of LTEX Templates January 12, 2022 ar X iv :2 10 9. 03 61 7v 6 [ m at h. G M ] 1 1 Ja n 20 22 With the ERP of a Kn+1-free graph G, one can obtain some usefull conclusion of the coloring of G.