Average Local Independence and the Spanning-Tree Leaf Number: A Proof of Graffiti.pc Conjecture 2

Abstract

We prove Graffiti.pc Conjecture 2, a 1996 conjecture listed as open on the \emph{Written on the Wall II} page marked ``Last update 7/23/26.'' Let $G$ be a finite simple connected graph. For $v\in V(G)$, let $I(v)=\hat{I}\pm (G[N_G(v)])$, and let $I_{\mathrm{avg}}(G)$ be the average of these local independence numbers. The conjecture states that the maximum number $L_s(G)$ of leaves in a spanning tree of $G$ satisfies $L_s(G)\ge 2(I_{\mathrm{avg}}(G)-1)$. We establish this inequality by extracting a triangle-free spanning subgraph that retains at least half of the total local-independence mass. A degree-square argument then produces a double star with sufficiently many leaves, and this tree extends to a spanning tree without losing leaves. Balanced complete bipartite graphs show that the bound is sharp.

5 pages, no figures; self-contained proof