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

Abstract

Let $A_n=(a_1,a_2,\dots ,a_n)$ and $B_n=(b_1,b_2,\dots ,b_n)$ be two sequences of nonnegative integers satisfying $a_1\ge a_2\ge \cdots \ge a_n$, $a_i\le b_i$ for $i=1,2,\dots ,n$, and $a_i=a_{i+1}$ implies that $b_i\ge b_{i+1}$ for $i=1,2,\dots ,n-1$. Let $I$ be a subset of $\{1,2,\dots ,n\}$ and $a_i\equiv b_i(\mathrm{mod}2)$ 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 $v_1,v_2,\dots ,v_n$, such that $a_i\le d_G(v_i)\le b_i$ for $i=1,2,\dots ,n$ and $d_G(v_i)\equiv b_i(\mathrm{mod}2)$ 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.