gpt pro 5.2 attempt on Math-5404

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that\[\sum\frac{1}{a_i}\gg \log k?\]Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph? A problem of Erdős and Hajnal. Gyárfás, Komlós, and Szemerédi [GKS84] have proved that this sum is $\gg \log k$, so that only the second question remains. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$. See also the entry in the graphs problem collection . See also [57] . References [GKS84] Gyárfás, A. and Komlós, J. and Szemerédi, E., On the distribution of cycle lengths in graphs . J. Graph Theory $1984$, 441-462. [LiMo20] Liu, Hong and Montgomery, Richard, A solution to Erdős and Hajnal's odd cycle problem . arXiv:2010.15802 $2020$.

FORMAL RESTATEMENT

Let $G$ be a finite simple graph with $n:=|V(G)|$ and $m:=|E(G)|=kn$ edges (so $k=m/n$ may be non-integer). Let $\mathcal{L}(G)=\{\ell\in\mathbb{N}: G \text{ contains a cycle of length }\ell\}$, and write its elements in increasing order $a_1<a_2<\cdots<a_t$. Define \[ S(G):=\sum_{i=1}^t \frac{1}{a_i}. \] Questions:

1. Does there exist an absolute constant $c>0$ such that for all $k\ge1$ and all graphs $G$ with $m=kn$ edges, \[ S(G)\ge c\log k?\]
2. For fixed $n$ and $m=kn$, is $S(G)$ minimized (among all graphs on $n$ vertices with $m$ edges) by a complete bipartite graph?

QUICK LITERATURE/CONTEXT CHECK

Only what is explicitly stated in the problem statement is used:

• The lower bound $S(G)\gg\log k$ is recorded as proved by Gyárfás--Komlós--Szemerédi.
• Liu--Montgomery are recorded as proving the asymptotically sharp bound $S(G)\ge (\tfrac12-o(1))\log k$.
• Therefore, according to the statement, the first question is settled and the second (minimizer) question remains.

No other external results are assumed.

ATTACK PLAN

• Since the first question is stated as solved in the problem statement, focus on the second question.
• Analyze $S(G)$ for complete bipartite graphs explicitly.
• Do exhaustive small-$n$ checks for instances where a complete bipartite graph exists with the same $(n,m)$, to see whether it minimizes $S(G)$ in those cases.

WORK

Lemma 1 (Cycle lengths and $S$ for complete bipartite graphs). Let $G=K_{a,b}$ with $a,b\ge 1$ and $n=a+b$. Then $G$ contains cycles of exactly the even lengths \[ 4,6,8,\dots,2\min\{a,b\}, \] and no odd cycles. Consequently, \[ S(K_{a,b})=\sum_{j=2}^{\min\{a,b\}} \frac{1}{2j}=\frac12\Big(H_{\min\{a,b\}}-1\Big), \] where $H_t:=\sum_{j=1}^t \frac1j$ is the $t$th harmonic number.

Proof. (Existence) Fix $2\le r\le \min\{a,b\}$. Choose distinct vertices $x_1,\dots,x_r$ in the $a$-part and distinct vertices $y_1,\dots,y_r$ in the $b$-part. Then the cycle \[ x_1-y_1-x_2-y_2-\cdots-x_r-y_r-x_1 \] uses $2r$ vertices and all consecutive pairs are adjacent in $K_{a,b}$, giving a cycle of length $2r$. Thus all even lengths $2r$ with $2\le r\le\min\{a,b\}$ occur.

(Nonexistence of odd cycles) $K_{a,b}$ is bipartite, so every cycle alternates between the two parts and hence has even length.

The formula for $S(K_{a,b})$ follows by summing $1/(2r)$ over $r=2,\dots,\min\{a,b\}$. $\square$

Lemma 2 (A dense subgraph with minimum degree $\ge k$). Let $G$ be a graph with $n$ vertices and $m=kn$ edges. Then $G$ contains a nonempty subgraph $H$ with minimum degree $\delta(H)\ge \lfloor k\rfloor$. Moreover, if $k$ is an integer, then $G$ contains a nonempty subgraph $H$ with $\delta(H)\ge k$.

Proof. Let $t:=\lfloor k\rfloor$. Repeatedly delete a vertex of degree at most $t-1$ (and all incident edges) as long as such a vertex exists. If the process deletes all vertices, then we can bound the original number of edges as follows: when a vertex is deleted, it has degree at most $t-1$ in the current graph, so we delete at most $t-1$ edges. Summing over all $n$ deletions gives \[ m\le (t-1)n < tn \le kn=m, \] which is impossible. Therefore the process stops with a nonempty remaining subgraph $H$ in which every vertex has degree at least $t$. That is, $\delta(H)\ge t=\lfloor k\rfloor$. If $k$ is integer, then $t=k$ and the same argument gives $\delta(H)\ge k$. $\square$

FAST REALITY CHECK (exact small instances for the minimizer question). For small $n$ where exhaustive search is feasible, I computed $\min S(G)$ over all graphs with fixed $(n,m)$ and compared it to the value for the complete bipartite graph with the same $(n,m)$ when such a $K_{a,b}$ exists. Exact results found:

• $(n,m)=(6,9)$: the complete bipartite graph is $K_{3,3}$. The minimum over all $6$-vertex, $9$-edge graphs is $S=\frac14+\frac16=\frac{5}{12}$, attained by $K_{3,3}$.
• $(n,m)=(7,10)$: the complete bipartite graph is $K_{2,5}$. The minimum is $S=\frac14$, attained by $K_{2,5}$ (its only cycle length is $4$).
• $(n,m)=(7,12)$: the complete bipartite graph is $K_{3,4}$. The minimum is $S=\frac14+\frac16=\frac{5}{12}$, attained by $K_{3,4}$.

These checks are consistent with the minimizer conjecture in these small cases but do not constitute a proof.

VERIFICATION

• Lemma 1: the construction gives a simple cycle of length $2r$ with all vertices distinct; bipartiteness rules out odd cycles.
• Lemma 2: the deletion argument is standard; the strict inequality $(t-1)n < kn$ uses $t=\lfloor k\rfloor$ and the fact $k\ge t$.
• Computation: small-$n$ exhaustive searches were performed by enumerating all graphs with the given number of edges and checking the set of cycle lengths by testing containment of all simple cycles in $K_n$ (cycle edge-sets) for lengths $3$ through $n$.

FINAL

UNRESOLVED

(i) Strongest proved partial result here: an explicit formula for $S(K_{a,b})$ in terms of $\min\{a,b\}$ (Lemma 1), and small-$n$ exhaustive verification that complete bipartite graphs minimize $S(G)$ for $(n,m)=(6,9),(7,10),(7,12)$.

(ii) First gap: prove (or disprove) that for every $(n,m)$ the minimum of $S(G)$ over all $n$-vertex graphs with $m$ edges is attained by a complete bipartite graph (when such a graph with exactly $m$ edges exists, or in an appropriate "nearest"/asymptotic sense).

(iii) Top 3 next moves:

• Try to show that any graph with $m$ edges either contains a short odd cycle (forcing a $1/3$ term) or else is close to bipartite, and then compare its even-cycle length spectrum to that of $K_{a,b}$.
• Extend exhaustive verification to larger $n$ using optimization/ILP to search for potential counterexamples to the minimizer claim.
• Seek a monotonicity/transfer principle: show that "spreading edges across a bipartition" decreases $S(G)$, formalizing why complete bipartite might be extremal.

(iv) Minimal counterexample structure (if the minimizer conjecture were false): a graph $G$ with the same $(n,m)$ as some $K_{a,b}$ but with strictly smaller $S(G)$, meaning it must avoid some cycle lengths present in $K_{a,b}$ (especially the short ones like $4$ and $6$) while still keeping $m$ edges.