A Variation of the Partial Parity \((g, f)\)-Factor Theorem Due to Kano and Matsuda

Let \( A_n = (a_1, a_2, \ldots, a_n) \) and \( B_n = (b_1, b_2, \ldots, b_n) \) be two sequences of nonnegative integers satisfying \( a_1 \geq a_2 \geq \cdots \geq a_n \), \( a_i \leq b_i \) for \( i = 1,2,\ldots,n \), and \( a_i = a_{i+1} \) implies that \( b_i \geq b_{i+1} \) for \( i = 1,2,\ldots,n-1 \). Let \( I \) be a subset of \( \{1,2,\ldots,n\} \) and \( a_i \equiv b_i \pmod{2} \) for each \( i \in I \). \( (A_n; B_n) \) is said to be partial parity graphic with respect to \( I \) if there exists a simple graph \( G \) with vertices \( v_1, v_2, \ldots, v_n \), such that \( a_i \leq d_G(v_i) \leq b_i \) for \( i = 1,2,\ldots,n \) and \( d_G(v_i) \equiv b_i \pmod{2} \) for each \( i \in I \). In this paper, we give a characterization for \( (A_n; B_n) \) to be partial parity graphic. This is a variation of the partial parity \( (g, f) \)-factor theorem due to Kano and Matsuda in degree sequences.