Degree sequences realizing labelled $h$-factors

Abstract

For a positive integer \( k \), let \( [k] = \{1, 2, \ldots, k\} \). Let \( h \) be a non-negative integer, and let \( n \) be a multiple of \( h + 1 \). Define \( H \) as the disjoint union of \( n/(h+1) \) cliques (each of size \( h + 1 \)) with vertex sets \( V_1, \ldots, V_{n/(h+1)} \), where \( V_i = \{ v_j \mid j = (i-1)(h+1) + k, k \in [h+1] \} \) for \( i \in [n/(h+1)] \). A non-increasing integer sequence \( (d_1, \ldots, d_n) \) is \( H \)-realizable if there exists a graph \( G \) with \( V(G) = V(H) = \{ v_i \mid i \in [n] \} \), \( d_G(v_i) = d_i \) for all \( i\in [n] \), and \( G \) contains \( H \) as a spanning subgraph. If \( h = 0 \), then a non-increasing integer sequence \( (d_1, \ldots, d_n) \) is \( H \)-realizable if and only if there exists a graph \( G \) with degree sequence \( (d_1, d_2, \dots, d_n) \); Erd\H{o}s and Gallai established a necessary and sufficient condition for this property. Recently, Briggs, McDonald, and Shan extended their result to the case \( h = 1 \). In this paper, we establish a necessary and sufficient condition for a sequence \( (d_1, d_2, \dots, d_n) \) to be \( H \)-realizable for any non-negative integer \( h \), thereby confirming a conjecture due to Briggs, McDonald and Shan.