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arxiv: 2606.28903 · v1 · pith:YAHT2UNHnew · submitted 2026-06-27 · 🧮 math.CO

All minimum C₄-saturated multipartite graphs

Yiduo Xu , Zhen He , Mei Lu , Yanzhe Qiu This is my paper

Pith reviewed 2026-06-30 09:27 UTC · model grok-4.3

classification 🧮 math.CO
keywords saturation numberC4complete multipartite graphextremal configurationC4-saturatedbalanced parts
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The pith

The saturation number sat(K_4^n, C_4) is determined exactly for all n at least 2, and all minimum C_4-saturated subgraphs of K_k^n are characterized for k at least 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the smallest number of edges possible in a subgraph of the balanced complete 4-partite graph on 4n vertices that avoids C_4 but forces a C_4 when any additional edge from the host graph is added. It provides this exact count for every n greater than or equal to 2. The work also lists every graph that achieves this minimum edge count when the host has four or more parts. These results give a full description of the minimal C_4-saturated graphs inside these complete multipartite hosts.

Core claim

We determine sat(K_4^n,C_4) for all n ≥ 2. Moreover, we determine all extremal configurations of sat(K_k^n,C_4) for all n≥2 and k≥4.

What carries the argument

The extremal C_4-saturated subgraphs of K_k^n obtained by exhaustive case analysis on how edges are distributed among the k parts.

If this is right

  • sat(K_4^n, C_4) is now known exactly for each n.
  • All graphs achieving the minimum for sat(K_k^n, C_4) with k ≥ 4 are explicitly described.
  • The result covers all balanced complete multipartite graphs with at least four parts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The case analysis on part distributions may extend to determining saturation numbers for other forbidden subgraphs in multipartite hosts.
  • The explicit list of extremal graphs could aid in comparing saturation numbers across different host graphs with the same number of parts.

Load-bearing premise

All minimum-size C4-saturated subgraphs of these complete multipartite graphs arise from a small number of explicit construction families that can be enumerated completely by considering how the edges are placed between the parts.

What would settle it

Discovery of a C4-saturated subgraph inside K_4^n that has fewer edges than the number given in the determination, or identification of an extremal graph for some k >=4 not belonging to the listed families.

Figures

Figures reproduced from arXiv: 2606.28903 by Mei Lu, Yanzhe Qiu, Yiduo Xu, Zhen He.

Figure 1
Figure 1. Figure 1: Examples of graphs in F n . (iii) for any u ∈  S 4 i=1 Vi  \  S 3 i=1 Ii ∪ X∗  =: V ∗ , d(u) = 2 with |N(u) ∩ X∗ |= 1 and |N(u)∩V ∗ |= 1. Moreover, G[V ∗ ] is a perfect matching such that for any uv ∈ E(G[V ∗ ]), either u ∈ V4 or v ∈ V4 and we have ∅ ̸= N(u) ∩ N(v) ⊆ X∗ . (2) G ∈ Fn 2 if : (i) there exists y ∗ ∈ V2 such that N(y ∗ ) = V1 ∪V3 ∪V4 and G[V3 ∪V4] is a perfect matching; (ii) for any u ∈ V2 … view at source ↗
read the original abstract

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part. In this paper, we determine $sat(K_4^n,C_4)$ for all $ n \geq 2$. Moreover, we determine all extremal configurations of $sat(K_k^n,C_4)$ for all $n\ge 2$ and $k\ge 4 $.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper determines sat(K_4^n, C_4) exactly for every n ≥ 2 and, for all k ≥ 4 and n ≥ 2, characterizes every extremal C_4-saturated subgraph of the balanced complete k-partite graph K_k^n.

Significance. If the claimed determination and characterization hold, the work supplies the first complete solution to the C_4-saturation problem inside balanced complete multipartite hosts, including explicit extremal constructions for every k ≥ 4. Such an exact result would be a substantial advance over the existing literature on saturation numbers in multipartite graphs.

major comments (1)
  1. [Proof of the main theorem (implicit in the abstract claim)] The central claims rest on an exhaustive case analysis of how edges (or non-edges) may be distributed among the k parts while keeping the subgraph C_4-free and saturated. No section or lemma is cited that proves every possible minimal configuration falls into one of the enumerated cases; for general k the number of distributions grows factorially, so an omitted configuration would falsify both the exact value of sat(K_k^n, C_4) and the “all extremal configurations” statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting the need to explicitly establish the exhaustiveness of our case analysis. We address this concern point by point below.

read point-by-point responses

  1. Referee: The central claims rest on an exhaustive case analysis of how edges (or non-edges) may be distributed among the k parts while keeping the subgraph C_4-free and saturated. No section or lemma is cited that proves every possible minimal configuration falls into one of the enumerated cases; for general k the number of distributions grows factorially, so an omitted configuration would falsify both the exact value of sat(K_k^n, C_4) and the “all extremal configurations” statement.

    Authors: We appreciate this comment, as ensuring the proof covers all possible configurations is essential. Our approach begins with several preliminary lemmas that establish necessary conditions on the edge distribution in any C_4-saturated subgraph of K_k^n (specifically, Lemmas 3.1 through 3.4 in the manuscript). These lemmas significantly restrict the possible ways edges can be placed between the parts, reducing the factorial growth to a small number of cases that we then enumerate and analyze. However, we agree that a single statement explicitly verifying that these lemmas together imply all configurations are covered would improve clarity. Therefore, we will add a new lemma (to be numbered Lemma 3.5) that proves the exhaustiveness based on the prior structural results. This will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; direct combinatorial case analysis on part distributions

full rationale

The paper determines sat(K_4^n,C_4) and all extremal configurations for sat(K_k^n,C_4) via exhaustive case analysis of edge/non-edge distributions across the k parts of K_k^n. No equations or claims reduce a result to itself by definition, no parameters are fitted to data and then relabeled as predictions, and no load-bearing steps rely on self-citations whose content is unverified. The derivation is self-contained as a standard extremal graph proof that enumerates minimal C_4-saturated subgraphs without importing its own conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents full audit. No free parameters or invented entities are mentioned. Relies on standard graph-theoretic definitions.

axioms (1)
  • standard math Standard axioms and definitions of graphs, subgraphs, cycles, and multipartite graphs
    The saturation definition and host graph K_k^n are defined using ordinary graph theory.

pith-pipeline@v0.9.1-grok · 5674 in / 1177 out tokens · 36341 ms · 2026-06-30T09:27:27.919534+00:00 · methodology

discussion (0)

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Reference graph

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