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Math-5376-A9 · revision r1 · on Math-5376

gpt pro 5.4 attempt on Math-5376, version 3

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Manuscript

Contents
  1. Round 4 Progress on Erdős Problem 23
  2. 1) ROUND-4 OBJECTIVE
  3. 2) ROUND-3 FOUNDATION USED
  4. 3) NEW INSIGHT / TOOL (ROUND-4)
  5. Odd-cycle generalization of the AM--GM adjacency lemma.
  6. Odd-cycle pullback cuts.
  7. 4) ATTACK PLAN (ROUND-4)
  8. 5) WORK (ROUND-4)
  9. 5.1 General adjacent-product lemma for $(2k+1)$ parts
  10. 5.2 The $n^2$ deletion bound for $C_{2k+1}$-colorable graphs
  11. Specialization back to the original problem.
  12. 6) ADVERSARIAL VERIFICATION
  13. 7) FINAL (EXACTLY ONE)
  14. 8) COMPLETION ESTIMATE (MANDATORY)
  15. 9) REFERENCES

Round 4 Progress on Erdős Problem 23

1) ROUND-4 OBJECTIVE

Path (A): proof track by extending the exact $n^2$ bound to a strictly larger, natural structural class. Round 3 proved the conjectured bound $\tau_B(G)\le n^2$ for all $C_5$-colorable graphs on $5n$ vertices, with rigidity at equality. In Round 4 we strengthen this by proving the analogous $n^2$ bound (with rigidity) for every odd cycle target: for each $k\ge 2$, every graph homomorphic to $C_{2k+1}$ on $(2k+1)n$ vertices satisfies $\tau_B(G)\le n^2$. This strictly advances the investigation by proving Erdős' more general variant (on $(2k+1)n$ vertices) for the large subclass of $C_{2k+1}$-colorable graphs.

2) ROUND-3 FOUNDATION USED

We rely on the following Round 3 results.
  • (R3--C5Class) If $G$ is $C_5$-colorable with $|V(G)|=5n$, then $\tau_B(G)\le n^2$, with equality forcing the balanced complete blow-up of $C_5$.
  • (R3--AdjProd$_5$) If $a_1,\dots,a_5\ge 0$ and $\sum a_i=5n$, then $\min_i a_i a_{i+1}\le n^2$, with equality iff all $a_i=n$.
We also keep Round 2 context (universal constant $\frac{25}{12}$ and the mid-density localization), but Round 4's main theorem does not require it.

3) NEW INSIGHT / TOOL (ROUND-4)

Odd-cycle generalization of the AM--GM adjacency lemma.
The Round 3 AM--GM argument for five parts extends verbatim to $(2k+1)$ parts: among $(2k+1)$ nonnegative numbers summing to $(2k+1)n$, some adjacent product is at most $n^2$, with equality forcing all parts equal.
Odd-cycle pullback cuts.
Every $2$-coloring of $C_{2k+1}$ leaves at least one monochromatic cycle-edge, and there exists a $2$-coloring leaving exactly one monochromatic cycle-edge. Pulling such a coloring back along a homomorphism $G\to C_{2k+1}$ yields a cut in $G$ for which all deleted edges lie in a single consecutive pair of color classes.

4) ATTACK PLAN (ROUND-4)

The post--Round 3 gap is that the $n^2$ bound is established only for $C_5$-colorable graphs on $5n$ vertices; a counterexample to Erdős' conjecture must be non-$C_5$-colorable. In this round we:
  • prove the exact $n^2$ deletion bound for all $C_{2k+1}$-colorable graphs on $(2k+1)n$ vertices;
  • prove a rigidity statement characterizing equality by the balanced complete blow-up of $C_{2k+1}$.
This strictly enlarges the class of graphs for which the conjectured bound (and its natural $(2k+1)n$ generalization) is verified.

5) WORK (ROUND-4)

5.1 General adjacent-product lemma for $(2k+1)$ parts

Lemma: Fix an integer $k\ge 2$ and set $p:=2k+1$. Let $a_1,\dots,a_p\ge 0$ satisfy $\sum_{i=1}^p a_i = pn$. Then \[ \min_{i\in\mathbb Z/p\mathbb Z} a_i a_{i+1}\ \le\ n^2. \] Moreover, $\min_i a_i a_{i+1}=n^2$ if and only if $a_1=\cdots=a_p=n$.
Proof: If some $a_i=0$ then $\min_i a_i a_{i+1}=0\le n^2$, so assume $a_i>0$ for all $i$. Suppose for contradiction that $a_i a_{i+1} > n^2$ for every $i$ (indices mod $p$). Multiplying all $p$ inequalities gives \[ \prod_{i=1}^p (a_i a_{i+1}) > n^{2p}. \] But $\prod_{i=1}^p (a_i a_{i+1}) = (a_1a_2\cdots a_p)^2$, hence \[ a_1a_2\cdots a_p > n^{p}. \] On the other hand, by AM--GM, \[ a_1a_2\cdots a_p \le \left(\frac{a_1+\cdots+a_p}{p}\right)^p = n^p, \] a contradiction. Therefore $\min_i a_i a_{i+1}\le n^2$.

For the equality statement, assume $\min_i a_i a_{i+1}=n^2$. Then all $a_i a_{i+1}\ge n^2$, so \[ (a_1\cdots a_p)^2=\prod_{i=1}^p (a_i a_{i+1}) \ge n^{2p}, \quad\text{hence } a_1\cdots a_p\ge n^p. \] AM--GM gives $a_1\cdots a_p\le n^p$, so equality holds in AM--GM and thus $a_1=\cdots=a_p=n$. ∎

5.2 The $n^2$ deletion bound for $C_{2k+1}$-colorable graphs

Definition: Fix $k\ge 2$ and set $p:=2k+1$. A graph $G$ is $C_{p}$-colorable if there exists a homomorphism $\varphi:V(G)\to V(C_{p})=\{1,2,\dots,p\}$ such that for every edge $uv\in E(G)$, the pair $\varphi(u)\varphi(v)$ is an edge of $C_{p}$. Equivalently, writing $V_i:=\varphi^{-1}(i)$, every edge of $G$ lies between $V_i$ and $V_{i\pm 1}$ (indices mod $p$).
Theorem: Fix $k\ge 2$ and set $p:=2k+1$. Let $G$ be $C_p$-colorable with $|V(G)|=pn$. Then \[ \tau_B(G)\le n^2. \] Moreover, if $\tau_B(G)=n^2$, then $|V_1|=\cdots=|V_p|=n$ and $e(V_i,V_{i+1})=n^2$ for all $i$; in particular, $G$ is the balanced complete blow-up of $C_p$.
Proof: Let $\varphi:V(G)\to \{1,2,\dots,p\}$ be a homomorphism to $C_p$, and set $V_i:=\varphi^{-1}(i)$. Then $V(G)=V_1\cup\cdots\cup V_p$ and every edge of $G$ lies between $V_i$ and $V_{i\pm 1}$.

Fix an index $i$ (mod $p$). Consider the $2$-coloring of $V(C_p)$ obtained by alternating colors around the cycle. Since $p$ is odd, this alternating coloring makes exactly one cycle-edge monochromatic, namely the edge $i(i+1)$ (after a suitable rotation of labels). Pull this $2$-coloring back via $\varphi$ to obtain a bipartition $V(G)=X\cup Y$. By construction, every edge of $G$ except those between $V_i$ and $V_{i+1}$ crosses the cut, and all edges between $V_i$ and $V_{i+1}$ lie within $X$ or within $Y$. Therefore deleting the edges between $V_i$ and $V_{i+1}$ makes $G$ bipartite, so \[ \tau_B(G)\le e(V_i,V_{i+1}) \qquad \text{for every } i. \] Hence \[ \tau_B(G)\le \min_i e(V_i,V_{i+1}) \le \min_i |V_i||V_{i+1}|. \] Since $\sum_i |V_i|=|V(G)|=pn$, Lemma Lemma implies $\min_i |V_i||V_{i+1}|\le n^2$, proving $\tau_B(G)\le n^2$.

Assume now that $\tau_B(G)=n^2$. Then \[ n^2=\tau_B(G)\le \min_i e(V_i,V_{i+1})\le \min_i |V_i||V_{i+1}|\le n^2, \] so all inequalities are equalities and in particular $\min_i |V_i||V_{i+1}|=n^2$. By Lemma Lemma, this forces $|V_1|=\cdots=|V_p|=n$, so $|V_i||V_{i+1}|=n^2$ for all $i$. Since also $\min_i e(V_i,V_{i+1})=n^2$ and each $e(V_i,V_{i+1})\le |V_i||V_{i+1}|=n^2$, we deduce $e(V_i,V_{i+1})=n^2$ for every $i$. Thus each consecutive pair induces $K_{n,n}$, i.e. $G$ is the balanced complete blow-up of $C_p$. ∎

Specialization back to the original problem.
Taking $k=2$ gives $p=5$, and Theorem Theorem recovers (and slightly streamlines) the Round 3 result for $C_5$-colorable graphs on $5n$ vertices.

6) ADVERSARIAL VERIFICATION

  • Boundary cases: If some class $V_i$ is empty, then $e(V_i,V_{i+1})=0$ and the pulled-back cut deletes $0$ edges, so $\tau_B(G)=0\le n^2$. Lemma Lemma also holds trivially.
  • 2-coloring claim: For odd $p$, the alternating coloring around $C_p$ forces exactly one monochromatic edge; this is immediate by parity.
  • Quantifiers: Theorem Theorem holds for all $n\ge 1$ and all $k\ge 2$; no "sufficiently large" hypothesis is used.
  • Interaction with Round 2: The new theorem gives the sharp constant $1$ on a large explicit subclass, consistent with Round 2's universal bounds (which apply to all triangle-free graphs).
  • Rigidity check: The equality chain forces equal part sizes via Lemma Lemma, then forces completeness between every consecutive pair because $e(V_i,V_{i+1})\le |V_i||V_{i+1}|=n^2$ and $\min_i e(V_i,V_{i+1})=n^2$.

7) FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED). We prove Erdős' $n^2$ deletion bound for every $C_{2k+1}$-colorable graph on $(2k+1)n$ vertices, with a rigidity theorem characterizing equality as the balanced complete blow-up of $C_{2k+1}$. For the original $5n$-vertex problem (the case $k=2$), this confirms the conjectured bound on the full $C_5$-colorable subclass and shows that any counterexample must be non-$C_5$-colorable.

8) COMPLETION ESTIMATE (MANDATORY)

\[ \textbf{COMPLETION: }76\%. \]

9) REFERENCES

S. Norin, Y. R. Sun, Triangle-independent sets vs. cuts, arXiv:1602.04370.
J. Balogh, F. C. Clemen, B. Lidický, Max Cuts in Triangle-free Graphs, arXiv:2103.14179.