University of Isfahan Transactions on Combinatorics 2251-8657 14 4 2025 12 01 Star-critical connected Ramsey numbers for 2-colorings of complete graphs 211 222 28651 10.22108/toc.2024.140839.2157 EN Monu Moun Department of Mathematics, Central University of Haryana, Haryana, India Jagjeet Jakhar Department of Mathematics, Central University of Haryana, Haryana, India. Mark Budden Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723 Journal Article 2024 02 28 This paper builds upon Sumner's work by further investigating the concept of connected Ramsey numbers, specifically focusing on star-critical connected Ramsey numbers. We obtain star-critical connected Ramsey numbers for several cases of trees versus complete graphs, stars versus stars, and paths versus paths. The connected Ramsey number for a star versus $K_3$ is also evaluated. Exact values are also obtained for the connected Ramsey numbers of $K_{1,n}$ versus $K_3$. This research explores the interplay between connectivity and graph coloring within the context of Ramsey theory. Ramsey number connected Ramsey number star-critical Ramsey number https://toc.ui.ac.ir/article_28651_1d475a3047b6054f8fd40fc12ec44034.pdf