A survey of face-antimagic evaluations of graphs

PASS

The construction is valid for $P(6,2)$: the labels are exactly $1,\dots,38$, matching $|V|+|E|+|F|=12+18+8$. Recomputing the face weights gives the six pentagonal weights $155,165,\dots,205$ and the two triangular weights $215,225$, so each face-size class has common difference $d=10$ as required. This gives a legitimate additional $d$-antimagic labeling of type $(1,1,1)$ for a generalized Petersen graph $P(n,2)$. I did not find an existing literature source giving this specific $d=10$ construction for $P(6,2)$.

Novelty assessment

TYPE1

Classification rationale: The accepted result is only a single explicit labeling for the small graph $P(6,2)$ with $d=10$. Even if new, it gives no infinite family, classification, or method beyond a finite certificate. It minimally answers an open-ended “find other values” problem and would not support a standalone combinatorics paper.

Literature check: I found no prior source giving this exact $d=10$ type $(1,1,1)$ face-antimagic labeling for $P(6,2)$, nor a stronger theorem implying it. The 2017 survey records known values from Bača–Jendrol’–Miller–Ryan for $P(n,2)$, including $d=1$, $d\in\{0,2,3\}$ in the $n\equiv2\pmod4$ case, and $d\in\{2,3,6,9\}$ in the $n\equiv0\pmod4$ case, then leaves other values as Open Problem 1. Searches of open-access metadata and web sources for “face-antimagic”, “d-antimagic”, “P(n,2)”, “P(6,2)”, and “generalized Petersen” found only unrelated labeling variants or the cited survey/background papers.

Citation: M. Bača et al., “A survey of face-antimagic evaluations of graphs,” Australas. J. Combin. 69(3) (2017), 382–393. Background: M. Bača, S. Jendrol’, M. Miller, J. Ryan, “Antimagic labelings of generalized Petersen graphs that are plane,” Ars Combin. 73 (2004), 115–128.