Algorithmic complexity of weakly semiregular partitioning and the representation number

A graph G is weakly semiregular if there are two numbers a, b, such that the degree of every vertex is a or b. The weakly semiregular number of a graph G, denoted by wr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether wr(G) = 2 for a given bipartite graph G with at most three numbers in its degree set is NP-complete. Among other results, for every tree T , we show that wr(T ) ≤ 2 log2 ∆(T ) + O(1), where ∆(T ) denotes the maximum degree of T . A graph G is a [d, d+ s]-graph if the degree of every vertex of G lies in the interval [d, d+ s]. A [d, d + 1]-graph is said to be semiregular. The semiregular number of a graph G, denoted by sr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a semiregular graph. We prove that the semiregular number of a tree T is ⌈ ) 2 ⌉. On the other hand, we show that determining whether sr(G) = 2 for a given bipartite graph G with ∆(G) ≤ 6 is NP-complete. In the second part of the work, we consider the representation number. A graph G has a representation modulo r if there exists an injective map l : V (G) → Zr such that vertices v and u are adjacent if and only if |l(u) − l(v)| is relatively prime to r. The representation number, denoted by rep(G), is the smallest r such that G has a representation modulo r. Narayan and Urick conjectured that the determination of rep(G) for an arbitrary graph G is a difficult problem [38]. In this work, we confirm this conjecture and show that if NP 6= P, then for any ǫ > 0, there is no polynomial time (1 − ǫ) 2 -approximation algorithm for the computation of representation number of regular graphs with n vertices.