mathematics Article Coverability of Graphs by Parity Regular Subgraphs Mirko Petruševski 1,† and Riste Škrekovski 2,3, * ,†   Citation: Petruševski, M.; Škrekovski, R. Coverability of Graphs by Parity Regular Subgraphs. Mathematics 2021, 9, 182. https:// doi.org/10.3390/math9020182 Received: 23 December 2020 Accepted: 15 January 2021 Published: 18 January 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Faculty of Mechanical Engineering, Ss. Cyril and Methodius University, 1000 Skopje, North Macedonia; mirko.petrushevski@mf.edu.mk 2 Faculty for Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia 3 Faculty of Information Studies, University of Ljubljana, 8000 Novo Mesto, Slovenia * Correspondence: riste.skrekovski@fmf.uni-lj.si † These authors contributed equally to this work. Abstract: A graph is even (resp. odd) if all its vertex degrees are even (resp. odd). We consider edge coverings by prescribed number of even and/or odd subgraphs. In view of the 8-Flow Theorem, a graph admits a covering by three even subgraphs if and only if it is bridgeless. Coverability by three odd subgraphs has been characterized recently [Petruševski, M.; Škrekovski, R. Coverability of graph by three odd subgraphs. J. Graph Theory 2019, 92, 304–321]. It is not hard to argue that every acyclic graph can be decomposed into two odd subgraphs, which implies that every graph admits a decomposition into two odd subgraphs and one even subgraph. Here, we prove that every 3-edge-connected graph is coverable by two even subgraphs and one odd subgraph. The result is sharp in terms of edge-connectivity. We also discuss coverability by more than three parity regular subgraphs, and prove that it can be efficiently decided whether a given instance of such covering exists. Moreover, we deduce here a polynomial time algorithm which determines whether a given set of edges extends to an odd subgraph. Finally, we share some thoughts on coverability by two subgraphs and conclude with two conjectures. Keywords: covering; even subgraph; odd subgraph; T-join; spanning tree 1. Introduction and Preliminaries We consider only undirected and finite graphs. Loops and/or parallel edges are allowed. Throughout, we use standard graph notation and terminology from [1]. There are only two types of graphs that are ‘parity regular’, that is, having all of their vertex degrees with the same parity. These are called ‘even graphs’ and ‘odd graphs’, where a graph is said to be even (or odd, respectively) if each of its vertex degrees is even (or odd, respectively). A covering (also called a cover) of given graph G is a family F of (not necessarily edge-disjoint) subgraphs of G, such that ⋃ F∈F E(F)= E(G); in the more restrictive case of edge-disjointness, F is said to be a decomposition. A fundamental process in mathematics is that of partitioning (resp. covering) a set of objects into (resp. by) classes according to certain rules. Graph theory deals with a situation where the rules translate to ‘simpler’ subgraphs. In this article we prove several results about graph coverings comprised of parity regular subgraphs. Let G =(V, E) be a graph. For an orientation D of E(G), the resulting digraph is denoted D(G), and for each vertex v ∈ V(G), E + (v) and E - (v) denote the sets of arcs in D(G) having their tails and heads, respectively, at v. An integer-valued mapping f with domain E(G) makes the ordered pair (D, f ) an integer flow of G if the equation  e∈E + (v) f (e)=  e∈E - (v) f (e) holds for each vertex v ∈ V(G). The support of f is the set supp( f )={e ∈ E(G)∶ f (e)≠ 0}, and if supp( f )= E(G) then the integer flow (D, f ) is said to be nowhere-zero. Given an integer k, (D, f ) is called a k-flow if  f (e) < k for each edge e ∈ E(G). Mathematics 2021, 9, 182. https://doi.org/10.3390/math9020182 https://www.mdpi.com/journal/mathematics