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Abstract:We study the fixed-density semi-inducibility profiles of the red-blue star $S_{2,1}$, which has one distinguished center, two red edges and one blue edge. For an $n$-vertex graph $G$, let $N(S_{2,1},G)$ be the number of injective labeled copies in which the two red edges of $S_{2,1}$ are mapped to edges of $G$ and its blue edge is mapped to a non-edge of $G$, that is, \begin{align*}
N(S_{2,1},G)=
\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)). \end{align*} For every fixed red edge density $\beta\in[0,1]$, we determine both extremal $S_{2,1}$-densities. On the upper side, we prove the missing low-density range and, together with the theorem of Balogh, Lidický, Mubayi, Pfender and Volec for $\beta\ge 1/4$, obtain the full four-branch profile predicted in their work. On the lower side, we show that the natural endpoint profile coming from the quasi-star and quasi-clique constructions is not universal; the correct minimum is given by a one-parameter three-class complement-split family. The proofs use a transfer argument with degree-square tie-breaking, reducing the extremal analysis to almost-regular, threshold and finite-staircase optimizations.

Submission history

From: Jinghua Deng [view email]
[v1] Mon, 22 Jun 2026 13:54:11 UTC (39 KB)