Subdivision of graphs in [Formula: see text]

Kristiana Wijaya; Edy Tri Baskoro; Hilda Assiyatun; Djoko Suprijanto

doi:10.1016/j.heliyon.2020.e03843

Subdivision of graphs in $R (m K_{2}, P_{4})$

Heliyon. 2020 Jun 12;6(6):e03843. doi: 10.1016/j.heliyon.2020.e03843. eCollection 2020 Jun.

Authors

Kristiana Wijaya¹, Edy Tri Baskoro², Hilda Assiyatun², Djoko Suprijanto²

Affiliations

¹ Graph, Combinatorics, and Algebra Research Group, Department of Mathematics, FMIPA, Universitas Jember, Jalan Kalimantan 37 Jember 68121, Indonesia.
² Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung 40132, Indonesia.

Abstract

For any graphs $F, G$ , and H, the notation $F \to (G, H)$ means that any red-blue coloring of all edges of F will contain either a red copy of G or a blue copy of H. The set $R (G, H)$ consists of all Ramsey $(G, H)$ -minimal graphs, namely all graphs F satisfying $F \to (G, H)$ but for each $e \in E (F)$ , $(F - e) ↛ (G, H)$ . In this paper, we propose a simple construction for creating new Ramsey minimal graphs from the previous known Ramsey minimal graphs (by subdivision operation). In particular, suppose $F \in R (m K_{2}, P_{4})$ and let $e \in E (F)$ be an edge contained in a cycle of F, we construct a new Ramsey minimal graph in $R ((m + 1) K_{2}, P_{4})$ from graph F by subdividing the edge e four times.

Keywords: Matching; Mathematics; Path; Ramsey minimal graphs; Red-blue coloring.