GPT 6 Astra Ultra attempt on Math-5745

Erdős Problem 577

Statement and scope.

Does every graph on $4k$ vertices with minimum degree at least $2k$ contain $k$ vertex-disjoint copies of $C_4$? Wang proved the affirmative answer. This attempt proves sharpness of the degree threshold, the bipartite case, and a general packing of $k/2-O(\sqrt k)$ cycles. The full packing theorem remains unresolved in this attempt.

A bound for a graph with no four-cycle.

If a graph $H$ of order $s$ has minimum degree $d$ and no $C_4$, fix a vertex $v$. The number of length-two paths starting at $v$, without returning immediately to $v$, is \[ \sum_{u\in N_H(v)}(d_H(u)-1)\geq d(d-1). \] Their endpoints must be distinct: two paths with the same endpoint and different middle vertices give a four-cycle. The endpoints lie among the $s-1$ vertices other than $v$, including possibly neighbours of $v$. Therefore \[ d(d-1)\leq s-1, \qquad d\leq\frac{1+\sqrt{4s-3}}2. \tag{1} \] This counting does not require triangle-freeness.

A guaranteed partial packing.

Choose a maximal collection of $r$ vertex-disjoint four-cycles in the graph of the question. The graph remaining after their vertices are removed is $C_4$-free and has \[ s=4k-4r,\qquad \delta(H)\geq2k-4r. \tag{2} \] If $s=0$, the desired full packing has already been found. If $r\geq k/2$, the weaker bound below holds immediately. Otherwise the lower bound in $2$ is positive; apply $1$, using $s\leq4k$, to obtain \[ 2k-4r\leq\frac12+2\sqrt k. \] Thus in all cases there exists a packing satisfying \[ r\geq\frac k2-\frac{\sqrt k}{2}-\frac18. \tag{3} \] Integer rounding may of course strengthen this bound. It gives a linear number of disjoint quadrilaterals, but does not reach a spanning packing.

The threshold cannot be decreased.

The complete bipartite graph $K_{2k-1,2k+1}$ has order $4k$ and minimum degree $2k-1$. Every four-cycle uses two vertices in each class. Hence at most \[ \left\lfloor\frac{2k-1}{2}\right\rfloor=k-1 \] can be vertex-disjoint. This proves the asserted degree condition, if sufficient, is sharp for every $k$.

The bipartite case and the first order.

If the graph in the question is itself bipartite with classes $A,B$, its minimum degree implies $|A|,|B|\geq2k$. Their sum is $4k$, so both have size exactly $2k$. Every vertex must be adjacent to the entire other class. Partition each class into pairs and match these pairs across the classes; each resulting $K_{2,2}$ is a four-cycle, giving the required $k$ disjoint copies.

For $k=1$, a graph on four vertices with minimum degree two has complement of maximum degree one. Its missing edges form a matching, extendible to a perfect matching. Removing that perfect matching from $K_4$ leaves a four-cycle contained in the original graph. This also proves the smallest case without a bipartiteness assumption.

Remaining gap and source.

A maximal packing can leave vertices uncovered, and $1$ only controls the size and degree of the leftover graph. To force $r=k$, one needs a replacement or augmentation argument involving the already chosen cycles. No such argument is proved here. The complete result is Hong Wang, Proof of the Erdős--Faudree Conjecture on Quadrilaterals, Graphs and Combinatorics 26 $2010$, 833--877, https://doi.org/10.1007/s00373-010-0948-3. Its proof is not used to fill the gap. Current statement: https://www.erdosproblems.com/577, checked 24 September 2026. No novelty or formal verification is claimed.

Completion rate estimate: 40%.