Some More Updates on an Annihilation Number Conjecture: Pros and Cons

If α(G) + μ(G) = | V| , then G= (V, E) is a König–Egerváry graph, where α(G) denotes the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If d₁≤ d₂≤ ⋯ ≤ d_n is the degree sequence of G, then the annihilation numbera(G) of G is the largest integer k such that ∑i=1kdi≤|E| (Pepper, Binding independence, Ph.D. Dissertation, University of Houston, 2004; Pepper, On the annihilation number of a graph, in: Recent Advances in Electrical Engineering: Proceedings of the 15th American Conference on Applied Mathematics, pp 217–220, 2009). A set A⊆ V satisfying ∑a∈Adeg(a)≤|E| is an annihilation set; if, in addition, deg(v)+∑a∈Adeg(a)>|E|, for every vertex v∈ V(G) - A, then A is a maximal annihilation set in G. In Larson and Pepper (Graphs with equal independence and annihilation numbers. Electron J Comb 18:180, 2011) it was conjectured that the following assertions are equivalent: (i) α(G) = a(G) ; (ii)G is a König–Egerváry graph and every maximum independent set is a maximal annihilating set. Recently, it turned out that the implication “(i)⟹ (ii)” was not true. A series of corresponding counterexamples can be found in Hiller (Counterexamples to the characterisation of graphs with equal independence and annihilation number. arXiv:2202.07529v1 [math.CO], 2022). In Levit and Mandrescu (On an annihilation number conjecture. Ars Math. Contemp. 18, 359–369, 2020), we presented an infinite family of non-bipartite König–Egerváry graphs that invalidate the “ (ii)⟹ (i)” part of this conjecture. In this paper, we provide two more infinite families of counterexamples, one consisting of trees and the other one comprising non-tree bipartite graphs. We also show that the above conjecture is true for trees with α(G) = 4 , disconnected non-bipartite König–Egerváry graphs with α(G) = 4 , and disconnected bipartite graphs with α(G) = 4 excluding the three following counterexamples: C₄∪ 2 K₂, Domino∪ K₂ and K_{3 , 3}- e.