2008/10/05 by Anders Sune Pedersen, Pedersen, Anders Sune, Bjarne Toft +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #math.CO
paper · pdf · doi:10.48550/arxiv.0810.0846
arxiv created 2008/10/05 · openalex publication_date 2008/10/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Conjecture of Hadwiger implies that the Hadwiger number h times the independence number α of a graph is at least the number of vertices n of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality, replacing the independence number α by 2α-1, that is, (2α-1)⋅ h ≥ n. In 2005 Kawarabayashi, Plummer and the second author published an improvement of the theorem, replacing 2α- 1 by 2α- 3/2 when α is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing 2α- 1 by 2α- 2 when α is at least 3. In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number α is equal to 2. The case α= 2 of Hadwiger's Conjecture was first pointed out by Mader as an interesting special case.