Interpolation theorems for a family of spanning subgraphs

Abstract

Let G be a graph with order p, size q and component number ω. For each i between p − ω and q, let $C_i (G)$ be the family of spanning i-edge subgraphs of G with exactly ω components. For an integer-valued graphical invariant φ if H → H ^′ is an adjacent edge transformation (AET) implies |φ(H)-φ(H^')|≤1 then φ is said to be continuous with respect to AET. Similarly define the continuity of φ with respect to simple edge transformation (SET). Let M _j(φ) and m _j(φ) be the invariants defined by $M_j (\varphi )(H) = \mathop {\max }\limits_{T \in C_j (H)} \varphi (T),m_j (\varphi )(H) = \mathop {\min }\limits_{T \in C_j (H)} \varphi (T)$. It is proved that both M _p−ω(φ) and m _p−ω(φ;) interpolate over $\mathcal{C}_i (G),p - \omega \leqslant i \leqslant q$, if φ is continuous with respect to AET, and that M _j(φ) and m _j(φ) interpolate over $\mathcal{C}_i (G),p - \omega \leqslant j \leqslant i \leqslant q$, if φ is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.