Alon-Tarsi Number of Some Regular Graphs
S. Prajnanaswaroopa
TL;DR
The paper analyzes the Alon-Tarsi number (ATN) of graphs, linking it to the exponents in the graph polynomial $P=\prod_{i<j}(x_i-x_j)$ and its role as an upper bound on the choice and online choice numbers. Using the Combinatorial Nullstellensatz and the Alon-Tarsi framework, it determines ATN values for several graph families, including complete multipartite graphs, line graphs of complete graphs of even order, and other regular graphs. The main technique involves orienting graphs in Eulerian ways and applying parity arguments on spanning Eulerian subdigraphs to satisfy the Alon-Tarsi condition and derive exact ATN values. Key results include exact ATN values for $K_{n,n}$, $K_{m,n}$ with even $n$ and divisibility, regular bipartite graphs, complete multipartite graphs, line graphs of $K_n$ with $n=4k$, and corollaries for the List Coloring Conjecture via total graphs, yielding concrete bounds on colorability and online colorability for these families.
Abstract
The Alon-Tarsi number of a polynomial is a parameter related to the exponents of its monomials. For graphs, their Alon-Tarsi number is the Alon-Tarsi number of their graph polynomials. As such, it provides an upper bound on their choice and online choice numbers. In this paper, we obtain the Alon-Tarsi number of some complete multipartite graphs, line graphs of some complete graphs of even order, and line graphs of some other regular graphs.