Booksize Conjecture in Extremal Graph Theory

• Booksize Conjecture is a principle in graph theory asserting that graphs just above the bipartite edge threshold must contain a large 'book' of triangles sharing a common edge.
• It integrates extremal graph theory and spectral analysis to quantify the relationship between triangle counts and book sizes via sharp bounds and phase transitions.
• The analysis employs combinatorial and spectral methods to reveal structural rigidity and supersaturation phenomena with implications for Turán-type problems.

A book graph of size $r$ ($B_r$) is the graph formed by $r$ triangles sharing a common edge (the "spine"), i.e., a $K_2$ plus $r$ vertices each joined to both endpoints of the spine. The booksize of a graph $G$, denoted $\mathrm{bk}(G)$, is the largest $r$ such that $B_r \subseteq G$. The Booksize Conjecture, originally posed by Erdős, connects the emergence of large books with extremal combinatorial properties of graphs, particularly in the regime just beyond the Turán threshold for triangles. The conjecture and its solution, along with subsequent generalizations and spectral refinements, reveal intricate phase transitions in graph structure, combinatorial rigidity, and spectrally forced subgraph saturation.

1. Classical Booksize Conjecture and Extremal Results

The seminal booksize conjecture formulated by Erdős (1962) asserts that any $n$-vertex graph $G$ with $e(G) = \lfloor n^2/4\rfloor + 1$ must contain a book of size at least $n/6$. More precisely, for all sufficiently large $n$, the extremal $B_{r+1}$-free graph is the complete balanced bipartite graph $T_{n,2} = K_{\lfloor n/2\rfloor, \lceil n/2\rceil}$, satisfying $e(T_{n,2}) = \lfloor n^2/4 \rfloor$ edges. Edwards and independently Khadžiivanov–Nikiforov completely resolved this conjecture, showing that for all $n \geq 6r$,

$$\mathrm{ex}(n,B_{r+1}) = e\left(T_{n,2}\right) = \left\lfloor \frac{n^2}{4} \right\rfloor.$$

Thus, graphs just beyond the bipartite threshold must contain either a large book or numerous triangles (Mubayi, 2010).

2. Quantitative Extremal Theorems for Books and Triangles

The interplay between the counts of books and triangles has been quantified through several classical and contemporary results:

• Rademacher's Theorem: Every $n$-vertex graph $G$ with $e(G) = \lfloor n^2/4\rfloor + 1$ must contain at least $\lfloor n/2 \rfloor$ triangles.
• Edwards's Theorem: Every $n$-vertex graph with $e(G) = \lfloor n^2/4 \rfloor + 1$ edges contains a book of size at least $\lfloor n/6 \rfloor$, with this bound being asymptotically best possible.

Recent advances interpolated these results:

• For $\alpha \in (1/2,1)$, if $G$ has $e(G) = \lfloor n^2/4\rfloor + 1$ and $b(G) < \alpha n/2$, then

$t(G) \geq \alpha(1-\alpha)\frac{n^2}{4} - o(n^2)$

where $t(G)$ is the number of triangles in $G$ (Mubayi, 2010).

For $\alpha \in (1/3, 1/2)$, a separate regime emerges: $$t(G) \geq \beta(\alpha) n^3 \quad \text{for some } \beta(\alpha) > 0,$$ with optimal $\beta(\alpha)$ conjectured to be $(\alpha(1-\alpha)^2)/16$.

3. Extremal Constructions and Phase Transitions

The extremal constructions that attain the bounds in these theorems highlight phase transitions in the underlying combinatorics:

• Sharp construction for $\alpha(1-\alpha)\frac{n^2}{4}$ triangles: Start with $K_{n/2,n/2}$, remove a vertex $v$, and reconnect $v$ to vertices so as to keep $b(G) < \alpha n/2$, showing that the main lower bound is best possible up to lower-order terms (Mubayi, 2010).
• Phase transition at $\alpha=1/2$ and $\alpha=1$: Regularity-type arguments show qualitatively different growth rates for triangle count when $b(G)$ is large or small.

4. Spectral Graph Theory and New Supersaturation Bounds

Spectral methods have provided major new insights regarding the prevalence of books in dense graphs:

• For any $B_{r+1}$-free graph $G$ with $m \geq (9r)^2$ edges,

$\rho(G) \leq \sqrt{m},$

with equality iff $G$ is complete bipartite. Consequently, any graph with spectral radius $\rho(G) > \sqrt{m}$ (a "Nosal graph") must contain a book of size at least $\mathrm{bk}(G) > \frac{1}{9}\sqrt{m}$, improving previous bounds by an order of magnitude (Zhai et al., 15 Jan 2026).

• For non-bipartite $B_{r+1}$-free graphs with $m \geq (240r)^2$,

$\rho^2 < m-1 + \frac{2}{\rho-1}$

unless $G$ is isomorphic to the special extremal construction $S^+_{m,s}$. This settles conjectures on spectral extremizers and shows "supersaturation": any graph exceeding the bound must contain many copies of $B_r$.

Table: Spectral Booksize Bounds

Theorem	Lower Bound on $\mathrm{bk}(G)$	Applies to
Li-Liu-Zhang	$\frac{1}{144}\sqrt{m}$	$B_{r+1}$-free, Nosal
Zhai–Li–Lou	$\frac{1}{9}\sqrt{m}$	$B_{r+1}$-free, Nosal
Supersatur.	$\gtrsim \frac{1}{240}\sqrt{m}$	Non-bipartite, spectral

The proofs leverage Perron–Frobenius theory and eigenvector-driven vertex partitioning, combined with forbidden subgraph structure (Zhai et al., 15 Jan 2026).

5. Book Graphs, Color-Criticality, and Related Turán-Type Problems

Books $B_{r+1}$ are $3$-chromatic color-critical graphs: the removal of the common spine edge reduces their chromatic number. Simonovits's color-critical edge theorem applies, yielding that for all large $n$, $\mathrm{ex}(n, B_{r+1}) = e(T_{n,2})$. The structure of extremal $B_{r+1}$-free graphs is uniquely bipartite for large $n$, unlike, for instance, the extremal constructions for triangles in non-bipartite settings.

In the non-bipartite regime, the extremal graphs shift:

• For $r=0$, $B_1 \cong C_3$, the extremal graphs are constructed by adding a triangle to a maximal bipartite subgraph.
• For $r=1$, the extremal family is parameterized by augmenting a triangle to a bipartite core in a controlled way.
• For $r \geq 2$, the extremal graph is unique: add a new vertex adjacent to $r$ vertices in each part of a balanced bipartite graph, inducing maximal edge count subject to non-bipartiteness and absence of $B_{r+1}$ (Miao et al., 18 Aug 2025).

6. Proof Strategies and Combinatorial Methodologies

Key proof tools include:

• Removal Lemma (Ruzsa–Szemerédi): Low triangle count implies one can delete $o(n^2)$ edges to make the graph triangle-free.
• Stability Theorems: Almost extremal triangle-free graphs must be close to bipartite, allowing a strong structural handle.
• Re-insertion and Double Counting: After approaching bipartiteness, edges are reinserted, and constraints on booksize yield lower bounds on triangle counts via careful averaging and double counting arguments.

Spectral arguments exploit eigenvector coordinates to induce vertex partitioning consistent with forbidden subgraph conditions (Mubayi, 2010, Zhai et al., 15 Jan 2026).

7. Open Problems and Directions

An outstanding open question is the optimal value of $\beta(\alpha)$ in the cubic triangle count regime for $\alpha \in (1/3,1/2)$ (Mubayi, 2010). Spectral bounds for booksize in terms of graph order, edge count, and forbidden subgraph structure continue to be refined (Zhai et al., 15 Jan 2026). Extensions to other color-critical or extremal graphs, as well as to hypergraph analogs, represent important ongoing directions in extremal and spectral combinatorics.

• "Books vs Triangles" (Mubayi, 2010)
• "Advances on two spectral conjectures regarding booksize of graphs" (Zhai et al., 15 Jan 2026)
• "Turán number of books in non-bipartite graphs" (Miao et al., 18 Aug 2025)