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Statement

Here $m(n)$ is the minimum number of edges of a pancyclic graph on $n$ vertices (p. 1).

Conjecture 1 (p. 3), as posed: "$m(n)<m(n+1)$ for all $n\ge3$"

The paper motivates it by Table 1, where $m(n)$ increases by at least $1$ and at most $2$ from $n$ to $n+1$ for $n\le37$ (p. 3), and reads it as saying that a pancyclic graph on $n+1$ vertices needs at least as many chords as a minimal pancyclic graph on $n$ vertices.

Source. S. Griffin, Minimal pancyclicity, arXiv:1312.0274v1 (1 December 2013; 6 pages), the only arXiv version; Conjecture 1 on p. 3. A preprint. The edition read is identified in the source digest.

Read depth. Claims checked: the conjecture and the sentences around it were read on the page image of p. 3.

Proof pointer

Open in the paper. Partial results: Corollary 2 (p. 5) proves $m(n-1)<m(n)$ when some minimal pancyclic graph on $n>6$ vertices has an arc of length at least $(n-1)/2$; Corollary 3 (p. 5) gives, when some minimal pancyclic graph on an even number $n$ of vertices has an arc of length $n/2-1$ in its Hamiltonian cycle, either $m(n-1)<m(n)$ or $m(n/2+2)\le m(n)+2-n/2$. The values of Table 1 satisfy the conjecture for $3\le n\le36$.

Dependencies

Table 1 as evidence.

Bears on

• Problem 1016: in the problem's notation the conjecture says $h(n+1)\ge h(n)$, that is, $h$ is nondecreasing; a question about the shape of $h$, not the asymptotic question the problem asks.