COMBINATORIAL ANALYSIS (MATRIX PROBLEMS, ORDER THEORY)

PASS

The $n=1,d=4$ counterexample is rigorous. The bounds force the three edge-weights to be exactly $9,13,17$, forcing weights $9$ and $17$ to use endpoint pairs $\{1,2\}$ and $\{4,5\}$ with edge labels $6$ and $8$. In $P_2\cup P_3$, the remaining edge must then involve the unused vertex label $3$, so it cannot have endpoint sum $6$, as required for weight $13$ with edge label $7$. Thus no such labeling exists for $(n,d)=(1,4)$, disproving the universal existence claim. No prior matching result was found in the attempted literature search.

Novelty assessment

TYPE1

Classification rationale: This is a genuinely elementary finite counterexample: $P_2\cup P_3$ is not super $(a,4)$-edge-antimagic total. It only refutes the universal reading of the open-problem entry at the boundary case $n=1$, and does not resolve the remaining $n\ge2$ cases or the $d=3$ case. It is not substantial enough for a standalone paper.

Literature check: I found no prior source explicitly recording this $n=1,d=4$ counterexample. Searches for variants of $nP_2\cup P_{n+2}$, $P_2\cup P_3$, “super $(a,d)$-edge-antimagic total labeling,” “disconnected graphs,” and related graph-labeling papers led back to the Dafik/Miller/Ryan/Bača line of work and open-problem listings, but not to a published statement covering this nonexistence result.

Citation: No prior citation for the counterexample found. Background source: Dafik, M. Miller, J. Ryan, M. Bača, “On super $(a,d)$-edge-antimagic total labeling of disconnected graphs,” Discrete Mathematics 309 (2009), 4909–4915, DOI: 10.1016/j.disc.2008.04.031.