The maximum number of edges in $C_{2k+1}^{-}$-free unbalanced signed graphs with bounded clique number

Abstract

Recently, the Turán problem for graphs with bounded clique number has attracted considerable attention. Since a graph can be regarded as a signed graph without negative edges, it is natural to extend the study of such Turán-type problems to signed graphs. With this motivation, we investigate the Turán-type problem for unbalanced $C_{2k+1}^{-}$-free signed graphs with bounded clique number. In fact, we establish a general result via the classical stability theorem. Specifically, for sufficiently large $n$ and a color-critical graph $F$, we determine the maximum number of edges among all $n$-vertex $C_{2k+1}^{-}$-free unbalanced signed graphs whose underlying graphs are $F$-free.