Spectral extrema for avoiding three or more vertex-disjoint 5-cycles

Determine, for every sufficiently large number of vertices n and each fixed integer t≥3, the spectral extremal graphs among n-vertex 1-planar graphs containing no t pairwise vertex-disjoint copies of the cycle C5.

Background

The paper determines the unique spectral extremal graphs for C5-free and 2C5-free 1-planar graphs. Its structural reduction shows that, under the relevant hypotheses, an extremal graph contains a spanning complete bipartite graph K_{2,n-2}, reducing the problem to the component structure of the remaining graph. The authors ask for the corresponding characterization when the forbidden configuration consists of t≥3 vertex-disjoint 5-cycles, a case not resolved by the paper.

References

For sufficiently large $n$, let $t\ge3$ be a fixed integer. Determine the spectral extremal graphs among $n$-vertex $1$-planar graphs without $t$ vertex-disjoint copies of $C_5$.

— Spectral extrema of 1-planar graphs with no short cycles or small cliques  (2608.24519 - Li et al., 25 Aug 2026) in Section 6, Concluding remarks, first Problem environment