Application of polynomial method to on-line list colouring of graphs

A graph is on-line chromatic choosable if its on-line choice number equals its chromatic number. In this paper, we consider on-line chromatic-choosable complete multi-partite graphs. Assume G is a complete k -partite graph. It is known that if the polynomial P ( G ) defined as P ( G ) = ∏ u < v , u v ∈ E ( x u − x v ) has one monomial ∏ v ∈ V x v φ ( v ) with φ ( v ) < k whose coefficient is nonzero, then G is on-line k -choosable. It is usually difficult to calculate the coefficient of a monomial in P ( G ) . For several classes of complete multi-partite graphs G , we introduce different polynomials Q ( G ) associated to G , such that Q ( G ) and P ( G ) have the same coefficient for those monomials of highest degree. However, the calculation of the coefficient of some monomials based on Q ( G ) is easier. Using this method, we prove the following graphs are on-line k -choosable: K ℓ + 1 , 1 ∗ ( ℓ − 1 ) , 2 ∗ ( k − ℓ ) , K s , t , 1 ∗ ( k − 2 ) (where ( s − 1 ) ( t − 1 ) ≤ 2 k − 3 ), K 3 ∗ 2 , 1 ∗ 2 , 2 ∗ ( k − 4 ) and K 4 , 3 , 1 ∗ 3 , 2 ∗ ( k − 5 ) . These results provide support for the on-line version of Ohba’s conjecture: graphs G with | V ( G ) | ≤ 2 χ ( G ) are on-line chromatic-choosable.