Tight minimum degree conditions for apex-outerplanar minors and subdivisions in graphs and digraphs

Mathematics > Combinatorics [Submitted on 18 Mar 2024 (v1), last revised 18 Jun 2026 (this version, v2)] Title:Tight minimum degree conditions for apex-outerplanar minors and subdivisions in graphs and digraphs View PDF HTML (experimental)Abstract:Motivated by Hadwiger's conjecture and related problems for list-coloring, we study graphs $H$ for which every graph with minimum degree at least $|V(H)|-1$ contains $H$ as a minor. We prove that a large class of apex-outerplanar graphs satisfies this property. Our result gives the first examples of such graphs whose vertex cover numbers are significantly larger than half of the number of its vertices, which breaks a barrier for attacking related coloring problems via extremal functions, and recovers all known such graphs that have arbitrarily large maximum degree. Our proof can be adapted to directed graphs to show that if $\vec H$ is the digraph obtained from a directed cycle or an in-arborescence by adding an apex source, then every digraph with minimum out-degree $|V(\vec H)|-1$ contains $\vec H$ as a subdivision or a butterfly minor respectively. These results provide the optimal upper bound for the chromatic number and dichromatic number of graphs and digraphs that do not contain the aforementioned graphs or digraphs as a minor, butterfly minor and a subdivision, respectively. Special cases of our results solve an open problem of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé and strengthen results of Gishboliner, Steiner and Szabó. Submission history From: Youngho Yoo [view email][v1] Mon, 18 Mar 2024 04:42:24 UTC (31 KB) [v2] Thu, 18 Jun 2026 02:27:29 UTC (30 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.