Generalization of Ramsey Number for Cycle with Pendant Edges

Jagjeet Jakhar, Monu Moun, Youngsoo Seol, Majeed Ahmad Yousif, Muhammad Amer Latif () and Pshtiwan Othman Mohammed ()
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Jagjeet Jakhar: Department of Mathematics, Central University of Haryana, Jant-Pali, Mahendergarh 123031, India
Monu Moun: Department of Mathematics, Central University of Haryana, Jant-Pali, Mahendergarh 123031, India
Youngsoo Seol: Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
Majeed Ahmad Yousif: Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq
Muhammad Amer Latif: Department of Mathematics, College of Sciences, King Faisal University, Hofuf 31982, Saudi Arabia
Pshtiwan Othman Mohammed: Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq

Mathematics, 2025, vol. 13, issue 7, 1-21

Abstract: This paper explores various Ramsey numbers associated with cycles with pendant edges, including the classical Ramsey number, the star-critical Ramsey number, the Gallai–Ramsey number, and the star-critical Gallai–Ramsey number. These Ramsey numbers play a crucial role in combinatorial mathematics, determining the minimum number of vertices required to guarantee specific monochromatic substructures. We establish upper and lower bounds for each of these numbers, providing new insights into their behavior for cycles with pendant edges—graphs formed by attaching additional edges to one or more vertices of a cycle. The results presented contribute to the broader understanding of Ramsey theory and serve as a foundation for future research on generalized Ramsey numbers in complex graph structures.

Keywords: Ramsey number; star-critical Ramsey number; Gallai–Ramsey number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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