The Irredundance-related Ramsey Numbers \(s(3,8)=21\) and \(w(3,8)=21\)

For a graph G and for non-negative integers p, q, and r, the triplet \ ( (p, q, r) \) is said to be an admissible triplet if \ (3p + 4q + 6r = |E (G) |\). If G admits a decomposition into p cycles of length 3, q cycles of length 4, and r cycles of length 6 for every admissible triplet \ ( (p, q, r) \), then we say that G has a \ (\C₃^{p, C₄^q, C₆^r\}\) -decomposition. In this paper, the necessary conditions for the existence of \ (\C₃^{p, C₄^q, C₆^r\}\) -decomposition of \ (K, ₌, ₍ (m n) \) are proved to be sufficient. This affirmatively answers the problem raised in Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. 197/198 (1999), 123-135. As a corollary, we deduce the main results of Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. , 197/198, 123-135 (1999) and Decompositions of complete tripartite graphs into cycles of lengths 3 and 6, Austral. J. Combin. , 73 (1), 220-241 (2019).