A study on parity signed graphs: The rna number

A signed graph (G,σ) on n vertices is called a parity signed graph if there is a bijection f:V(G)→{1,2,…,n} such that for each edge e=uv of G, f(u) and f(v) have the same parity if σ(e)=1, and opposite parities if σ(e)=−1. The signature σ is called a parity-signature of G. Let Σ−(G) denote the set of the number of negative edges of (G,σ) over all possible parity-signatures σ. The rna number σ−(G) of G is given by σ−(G)=minΣ−(G). In other words, σ−(G) is minimum cardinality of an edge-cut of G such that the number of vertices in the two sides differ at most one. In this paper, we prove that for any graph G, Σ−(G)={σ−(G)} if and only if G is K1,n−1 with n even or Kn. This confirms a conjecture proposed by Acharya and Kureethara (2021)[1]. Moreover, we prove nontrivial upper bounds for the rna number: for any graph G on m edges and n (n≥2) vertices, σ−(G)≤⌊m2(1+12⌈n2⌉−1)⌋≤⌊m2+n4⌋. We show that Kn, Kn−e, and Kn−▵ are the only graphs reaching the bound ⌊m2+n4⌋. Finally, we prove that for any graph G, σ−(G)+σ−(G¯)≤σ−(G∪G¯), where G¯ is the complement of G. This solves a problem proposed by Acharya et al. (2021)[2].