Best monotone degree conditions for binding number

Abstract

a b s t r a c tWe give sufficient conditions on the vertex degrees of a graph G to guarantee that G has binding number at least b, for any given b > 0. Our conditions are best possible in exactly the same way that Chvátal's well-known degree condition to guarantee a graph is Hamiltonian is best possible.

Best Monotone Degree Conditions for Graph Properties: A Survey

“…3) As noted in Section 3.2, the bound bind(G) ≥ 3/2 in Theorem 3.2.2 is best possible, and thus there is no structural implication of the form b-binding ⇒ hamiltonian, for any b < 3/2. But this implication does hold in a best monotone sense for b > 1 [10]. then π ∈ BM(1-binding) by Theorem 3.2.5, but π ∈ BM(hamiltonian), since π fails to satisfy Theorem 1.1 for i = 1 2 n − 1.…”

“…In [10], a best monotone condition was given for a graph to be b-binding, first for 0 < b ≤ 1 and then for b ≥ 1.…”

“…But the converse is true in a best monotone sense, i.e., BM(χ(G) ≤ 2k) ⇒ BM(a(G) ≤ k), by Theorem 3.6.As noted in Section 3.2, the bound bind(G) ≥ 3/2 in Theorem 3.2.2 is best possible, and thus there is no structural implication of the form b-binding ⇒ hamiltonian, for any b < 3/2. But this implication does hold in a best monotone sense for b > 1[10]. Bauer et al[10]).…”

“…But this implication does hold in a best monotone sense for b > 1[10]. Bauer et al[10]). If b > 1, then BM(b-binding) ⇒ BM(hamiltonian).The hypothesis b > 1 in Theorem 5.2 is best possible: If π = then π ∈ BM(1-binding) by Theorem 3.2.5, but π ∈ BM(hamiltonian), since π fails to satisfy Theorem 1.1 for i = 1 2 n − 1.…”

Binding Number, $k$-Factor and Spectral Radius of Graphs

“…One issue concerning binding numbers involves characterizing its boundary by using other structural parameters of the graph, such as, degree sequence [3], minimum degree [4] and connectivity [30]. Similar to vertex-connectivity and edge-connectivity, toughness and binding number are measures of the vulnerability of a graph.…”