Fixed-density profiles for the semi-induced 4-vertex star

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 $β\in[0,1]$, 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 $β\ge 1/4$, 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.

• The paper completes the upper profile for S₂,₁ across every edge density, confirming a four-branch conjecture with a switching point β* = t*² defined by an explicit quintic equation.
• The paper refutes the proposed endpoint-only lower profile using a three-class construction at β = 9/10 and replaces it with the exact variational formula i(S₂,₁,β) = min Cβ(z).
• The proofs combine degree-based edge-transfer identities, degree-square tie-breaking, threshold-graph structure, and second-variation arguments without relying on flag algebras or graphon compactness.

The fixed-density semi-inducibility problem for $S_{2,1}$

This paper determines the complete fixed-density semi-inducibility profiles of the red-blue star $S_{2,1}$, a four-vertex pattern consisting of one distinguished center with two ordered red leaves and one blue leaf. For an $n$-vertex graph $G$ viewed as a red-blue coloring of its edges and non-edges, the semi-induced count is

$$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$

a purely degree-based functional. Writing $\beta_G = 2e(G)/n^2$, the paper studies

$$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$

over all graph sequences $(G_n)$ with red density tending to $\beta$. This fixed-density formulation answers Problem 9.3 of Basit–Granet–Horsley–Kündgen–Staden in this instance and fits the feasible-region framework of Liu–Mubayi.

Prior work by Balogh, Lidický, Mubayi, Pfender and Volec established the high-density upper profile ($\beta \ge 1/4$: value $S_{2,1}$0 on $S_{2,1}$1, $S_{2,1}$2 on $S_{2,1}$3) and conjectured both the low-density upper profile and an endpoint-based lower profile built from quasi-star and quasi-clique constructions. The present work resolves the first conjecture completely and refutes the second.

The upper profile: completing the four-branch conjecture

The main upper result states that for $S_{2,1}$4, where $S_{2,1}$5 is the unique root of $S_{2,1}$6,

$S_{2,1}$7

where $S_{2,1}$8 is a one-parameter maximum over split constructions (clique plus independent class complete to it plus isolates). Combined with the known dense-range theorem, this yields the full four-branch profile conjectured by Balogh et al., confirming their prediction including the existence of the switching point $S_{2,1}$9. A technical appendix proves that $n$0 exactly below $n$1 via a discriminant computation tied to the quintic defining $n$2.

The proof rests on a structural dichotomy for "chosen" extremal graphs (maximizers of $n$3 among graphs with given $n$4, tie-broken by maximizing the degree-square sum $n$5): every chosen upper-extremal graph is either a threshold graph or of the form $n$6 with almost-regular core $n$7 and isolated vertices $n$8. The dichotomy follows from edge-transfer rules derived from the exact identity

$n$9

for moving an edge from $G$0 to $G$1, together with the degree-square change $G$2. The threshold branch at density $G$3 reduces to a convex optimization over capped simplices of tail-neighborhood profiles; interior maxima are excluded by a Lagrange multiplier argument producing a polynomial certificate $G$4, while boundary faces collapse to split constructions. The almost-regular branch is handled directly: monotonicity of $G$5 on $G$6 caps its contribution at $G$7.

The lower profile: refuting the endpoint conjecture

The lower-profile conjecture of Balogh et al. asserted that $G$8 equals the minimum of the two endpoint values $G$9 (quasi-star) and $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$0 (quasi-clique). The paper disproves this with an explicit counterexample at $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$1: a three-class construction with class sizes $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$2, $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$3, $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$4 achieves normalized density $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$5, strictly below the conjectured minimum $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$6. This is a concrete numerical refutation, not merely a plausibility argument.

The correct answer is a one-parameter family: with $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$7 and $$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$8,

$$N(S_{2,1},G)=\sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)),$$9

The extremal construction has a clique $\beta_G = 2e(G)/n^2$0, with $\beta_G = 2e(G)/n^2$1 complete to an independent class $\beta_G = 2e(G)/n^2$2 and $\beta_G = 2e(G)/n^2$3 anti-complete to $\beta_G = 2e(G)/n^2$4; vertices of $\beta_G = 2e(G)/n^2$5 (degree $\beta_G = 2e(G)/n^2$6) contribute zero, so only classes $\beta_G = 2e(G)/n^2$7 and $\beta_G = 2e(G)/n^2$8 matter. Both endpoint constructions arise as degenerations ($\beta_G = 2e(G)/n^2$9 recovers the quasi-clique value $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$0; $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$1 recovers the quasi-star value $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$2), so the new family strictly generalizes them.

The lower-bound proof again uses chosen lower-extremal graphs (minimizing $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$3, tie-broken by maximizing $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$4) and a complementary dichotomy: either the high-degree set satisfies $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$5, or the graph is threshold. In the first case, a chord inequality bounds the normalized degree-square sum by $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$6, and Cauchy's inequality applied to the vectors $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$7 and $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$8 yields $$I(S_{2,1},\beta)=\sup p(S_{2,1},(G_n)), \qquad i(S_{2,1},\beta)=\inf p(S_{2,1},(G_n))$$9. In the threshold case, twin-class structure converts the graph into a weak closed staircase pattern $(G_n)$0 with $(G_n)$1 and $(G_n)$2, where $(G_n)$3. The finite-dimensional inequality $(G_n)$4 is proved by minimal-counterexample compactness: zero-gap deletion preserves both functionals, and interior local minima are excluded case-by-case — patterns with at most three gaps by direct calculus, four and five gaps by explicit edge-preserving second-variation moves along constraint curves, and six or more gaps by a clean anti-diagonal transfer move whose second derivative is $(G_n)$5 whenever $(G_n)$6, which the first-variation balance equations force.

Methodological observations

Two features of the method deserve note. First, the degree-square tie-breaking is essential on both sides: it upgrades weak transfer inequalities (where the $(G_n)$7-count is merely non-decreasing under a move) into strict contradictions, which is what forces the threshold/almost-regular dichotomies. Second, because $(G_n)$8 depends only on the degree sequence, the entire analysis is effectively one-dimensional in degrees; the threshold staircase normalization makes this explicit, with errors uniformly $(G_n)$9 independent of the number of twin classes. No flag algebra or graphon compactness machinery is required.

Limitations and open questions

The results are specific to $\beta$0, and several aspects do not immediately generalize. The upper-profile proof relies on the degree-only form of the objective; patterns whose semi-induced count depends on adjacency structure beyond degrees would require different tools. The switching point $\beta$1 is defined implicitly as the root of a quintic, and no closed form or independent characterization of $\beta$2 is given beyond the discriminant argument in the appendix. The lower-profile formula leaves open the purely analytic question of locating, for each $\beta$3, which $\beta$4 attains $\beta$5; the paper establishes existence and continuity but does not classify the minimizing branches or the transition points between them. Finally, whether analogous complement-split families improve the lower profiles of other red-blue stars or alternating paths remains unaddressed.

Conclusion

The paper completes the fixed-density upper semi-inducibility profile of $\beta$6 across all $\beta$7, confirming the four-branch conjecture of Balogh, Lidický, Mubayi, Pfender and Volec with the switching point identified as $\beta$8 for an explicitly defined algebraic constant $\beta$9. It simultaneously replaces the conjectured quasi-star/quasi-clique lower profile with a strictly larger three-class complement-split family, providing an explicit counterexample at $\beta \ge 1/4$0 and an exact variational formula $\beta \ge 1/4$1. The proofs combine edge-transfer arguments with degree-square tie-breaking, threshold-graph staircases, and finite-dimensional second-variation analysis, giving a self-contained treatment that avoids heavier graph-limit machinery.