Dense Eulerian graphs are $(1, 3)$-choosable

Abstract

A graph G is total weight (k, k )-choosable if for any total list assignment L which assigns to each vertex v a set L(v) of k real numbers, and each edge e a set L(e) of k real numbers, there is a proper total L-weighting, i.e., a mapping f :. This paper proves that if G decomposes into complete graphs of odd order, then G is total weight (1, 3)-choosable. As a consequence, every Eulerian graph G of large order and with minimum degree at least 0.91|V (G)| is total weight (1, 3)-choosable. We also prove that any graph G with minimum degree at least 0.999|V (G)| is total weight (1, 4)-choosable.