2021/09/02 by Huajing Lu, Lu, Huajing, Xuding Zhu +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2109.00792
openalex publication_date 2021/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: V(G) ∪ E(G) → ℝ such that for each z ∈ V(G) ∪ E(G), f(z) ∈ L(z), and for each edge uv of G, ∑e ∈ E(u)f(e)+f(u) ≠ ∑e ∈ E(v)f(e) + f(v). 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.