The Turán number of the Cartesian product of a star and an edge

Abstract: Let $C_k$ denote the cycle of length $k$, $S_t$ be a star with $t$ edges. And let $B_t$ be the graph consisting of $t$ copies of $C_4$ sharing one fixed edge. Equivalently, $B_t=K_2 \mathbin{\square} S_t$, which is the Cartesian product of a star with $t$ edges and an edge. Recently, Gao, Janzer, Liu and Xu [\textit{Israel J. Math. 269(2025)}] proved that the Turán number of $K_2\mathbin{\square} C_{2l}$ is $Θ(n<sup>{\frac{3}{2}})$ for every $l\ge 4$. In this paper, we obtain upper and lower estimates for the Turán number of $B_t$ in both the general and bipartite settings for every $t\geq 2$. For the lower bound, we use random construction based on the extremal structure of $C_4$. These results imply that $$\frac{1}{2\sqrt{2}}\leq \lim_{t\to \infty} \frac{\mathrm{ex}(n,B_t)}{\sqrt{t}}\leq \frac{1}{2}$$, and $$\frac{1}{4}\leq \lim_{t\to \infty} \frac{\mathrm{ex}_{bip}(n,B_t)}{\sqrt{t}}\leq \frac{1}{2\sqrt{2}}.$$ In the case of $B_2$, we obtain sharper estimates. We show that the Turán number of $B_2$ is approximately between $(0.518+o(1))n<sup>{\frac{3}{2}}$ and $(0.603+o(1))n<sup>{\frac{3}{2}}$. And in the bipartite setting, it is approximately between $(0.385+o(1))n<sup>{\frac{3}{2}}$ and $(0.468+o(1))n<sup>{\frac{3}{2}}$. Moreover, in the bipartite setting, we give a more general result, which shows that for every tree $T$ with $t$ edges, the bipartite Turán number of $K_2\mathbin{\square}T$ is at most $$\frac{\sqrt{t}}{2\sqrt{2}}(1+o(1))n<sup>{\frac{3}{2}}$$.

• The paper establishes sharp upper and lower bounds for ex(n, Bₜ), pinpointing the threshold for excluding multiple C₄'s sharing an edge.
• It employs probabilistic blow-up constructions and intricate double-counting methods to obtain improved constants, especially for t=2.
• The results extend to bipartite settings and generalize to Cartesian products with arbitrary trees, offering insights for extremal graph constructions.

The Turán Number of the Cartesian Product of a Star and an Edge

Introduction

This paper investigates the Turán number for the Cartesian product of a star $S_t$ with $t$ edges and an edge $K_2$, denoted as $B_t = S_t \mathbin{\square} K_2$, equivalently the graph consisting of $t$ copies of $C_4$ sharing a single edge (see Figure 1 below). Determining Turán numbers—maximum sizes of graphs avoiding a given subgraph—remains a central problem in extremal combinatorics, especially in the bipartite case, where the problem is notably challenging. The main contributions of this work are sharp upper and lower bounds for $\mathrm{ex}(n, B_t)$ (and its bipartite variant $\mathrm{ex}_{bip}(n, B_t)$), explicit determination of leading constants for small $t$ (notably $t=2$), and generalizations to Cartesian products of $t$0 with arbitrary trees.

(Figure 1)

Figure 1: The graph $t$1. The graph is comprised of $t$2 copies of $t$3 sharing a single edge.

Turán Numbers for $t$4: Upper and Lower Bounds

The main theorem establishes that for any $t$5 and sufficiently large $t$6,

$t$7

The lower bound, based on random and blow-up constructions from $t$8-free graphs, matches the upper bound in order of magnitude,

$t$9

with a more complex expression for even $K_2$0. As $K_2$1, this shows $K_2$2 is the window for the rescaled limit $K_2$3.

Improved Coefficients for $K_2$4

For $K_2$5, sharper analysis gives

$K_2$6

The upper bound comes via intricate double-counting and star-decomposition arguments leveraging exclusion of special structures, and it is improved further from the standard $K_2$7 to $K_2$8. The lower bound is achieved by randomized blow-ups maximizing edge count while precluding $K_2$9.

Figure 2: The red edge represents the edge shared by $B_t = S_t \mathbin{\square} K_2$0 copies of $B_t = S_t \mathbin{\square} K_2$1.

Bipartite Turán Numbers and Generalizations

The bipartite Turán number $B_t = S_t \mathbin{\square} K_2$2 is analyzed via similar methods but reflects tighter constraints:

$B_t = S_t \mathbin{\square} K_2$3

For the case $B_t = S_t \mathbin{\square} K_2$4, the result is further refined:

$B_t = S_t \mathbin{\square} K_2$5

This uses an adaptive deletion process and careful bounding of the number of forbidden configurations using degree distributions and codegree constraints.

A general theorem extends the upper bound to all trees $B_t = S_t \mathbin{\square} K_2$6 with $B_t = S_t \mathbin{\square} K_2$7 edges:

$B_t = S_t \mathbin{\square} K_2$8

Figure 3: For each $B_t = S_t \mathbin{\square} K_2$9, if $t$0 and $t$1, a forbidden $t$2-configuration is found via the neighborhoods (star structure is violated).

Figure 4: For $t$3, if $t$4, then a $t$5 can be constructed; no such overlap can occur in a $t$6-free graph.

Structural and Methodological Insights

A central theme is counting copies of $t$7 in the host graph via various projections (e.g., fixing a “good” set of size $t$8 with large common neighborhood) and using intricate double-counting strategies. Forbidding $t$9-configurations imposes rigid structural limits that are exploited via extremal combinatorial arguments, such as bounding the number of stars or matchings in associated auxiliary graphs.

The approach uses:

• Blow-up construction: Enlarging $C_4$0-free graphs by substituting each vertex with an independent set, maximizing edge count while preserving $C_4$1-freeness.
• Auxiliary graphs: Constructing graphs (e.g., $C_4$2 induced by pairs of neighborhoods) whose matching size or star-structure is tightly controlled due to $C_4$3 exclusion.
• Decomposition by degree: Deletion patterns based on low-degree vertices to reduce analysis to graphs with "large minimum degree", followed by analytic optimization over $C_4$4.
• Probabilistic tools: Lower bounds leveraging randomized partitions, optimizing over blow-up parameters.

Theoretical and Practical Implications

On the theoretical side, these results precisely quantify the extremal threshold for the exclusion of multiple $C_4$5's joined along a common edge. The approach provides an explicit route to determining leading constants in the $C_4$6 regime for a wide class of bipartite, even structured, graphs, extending earlier work from simple cycles and grids to star-propagated Cartesian products.

The results improve previously best-known constants in several key cases and establish, for the first time, exact asymptotics up to multiplicative constants for the bipartite Turán number of $C_4$7 and general $C_4$8. From a practical viewpoint, these bounds govern the extremal behavior of large, sparse bipartite graphs with prescribed forbidden substructures—relevant, for example, in design of extremal graph instances for network coding, pseudorandomness, and information theory.

Directions for Further Research

This work opens several questions for further investigation. Potential directions include:

• Extending the methodology to more general graphs of the form $C_4$9 for arbitrary $\mathrm{ex}(n, B_t)$0.
• Tightening the constants or giving exact bounds for more specific $\mathrm{ex}(n, B_t)$1, especially in the small and moderate regime.
• Investigating stability: characterizing extremal graphs realizing these numbers.
• Studying analogous problems in directed graphs or hypergraphs.
• Exploring applications in computational extremal graph theory, such as faster detection algorithms or extremal certificate construction.

Conclusion

This paper provides a comprehensive determination of Turán numbers for the Cartesian product $\mathrm{ex}(n, B_t)$2, giving matching upper and lower bounds up to tight constants and refining these constants in special cases. The analysis robustly generalizes to bipartite settings and to $\mathrm{ex}(n, B_t)$3 for trees $\mathrm{ex}(n, B_t)$4. The combination of probabilistic, combinatorial, and analytic tools yields an advanced framework for extremal problems concerning blow-ups and products of bipartite graphs, and establishes new benchmarks in this domain.

[The Turán number of the Cartesian product of a star and an edge, (2604.11366)]