Exact rainbow saturation numbers for generalized friendship graphs

Determine the exact value of rsat(n,F_{t,p,q}) for all sufficiently large n, for every fixed integers t\geq2, p\geq2, and q\geq1, where F_{t,p,q}=tK_p\vee K_q.

Background

Theorem 1.3 (Theorem TFriend) determines the asymptotic order and leading term for generalized friendship graphs F_{t,p,q}=tK_p\vee K_q, proving rsat(n,F_{t,p,q})=(p+q-1)n+O(1) for fixed t\geq2, p\geq2, and q\geq1. The paper explicitly leaves unresolved the determination of the exact value, rather than merely its asymptotic form, for all sufficiently large n.

References

For generalized friendship graphs, Theorem~\ref{TFriend} determines the asymptotic behavior and leaves a more concrete exact problem. For fixed integers $t\geq2$, $p\geq2$, and $q\geq1$, determine the exact value of $rsat(n,F_{t,p,q})$ for all sufficiently large $n$.

— On rainbow saturated graphs with minimum number of edges  (2609.34898 - Qiu et al., 28 Sep 2026) in Problem 5.2 (labelled prob:friendship), Section 5, Concluding remarks

Theorem~\ref{TFriend} assumes $p\geq2$ and therefore does not cover the case

F_{t,1,q}=(tK_1)\vee K_q,

where a clique $K_q$ is joined to an independent set of size $t$. When $q=1$, this graph is a star, which is discussed in Remark~\ref{R24}. This leads to the following question for $q\geq2$. For fixed integers $t\geq2$ and $q\geq2$, determine

rsat(n,F_{t,1,q}).

— On rainbow saturated graphs with minimum number of edges  (2609.34898 - Qiu et al., 28 Sep 2026) in Problem 5.3 (labelled prob:friendship-p-one), Section 5, Concluding remarks