Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs

Abstract

For a graph H, let σ 2 (H ) = min{d (u) + d (v ) | for every edge uv ∈ E (H )}. For r > 0 and k ∈ {2, 3}, let Q 0 (r, k ) be a set of k-edgeconnected K 3 -free graphs of order at most r and without spanning closed trails. We show that for given p > 0 and , if H is a k-connected claw-free graph of order n with δ(H ) ≥ 3 and σ 2 (H ) ≥ (2n + )/p, and if n is sufficiently large, then either H is Hamiltonian or the Ryjácȇk's closure cl (H ) = L(G) where G is an essentially k-edge-connected K 3 -free graph that can be contracted to a graph in Q 0 (5p − 10, k ). As applications, we prove:and n is sufficiently large, then H is Hamiltonian.(ii) For k = 3, if σ 2 (H ) > 2n+10 10 and n is sufficiently large, then H is Hamiltonian.These bounds are sharp. Furthermore, since the graphs in Q 0 (5p − 10, k ) are fixed for given p and can be determined in a constant time, any improvement to (i) or (ii) by increasing the value of p and so enlarging the number of exceptions can be obtained computationally.

Degree conditions for Hamiltonian properties of claw-free graphs

“…However, if we consider claw-free graphs, this picture changes. This was first observed by Chen [34] who considered the Brualdi-Shanny condition for guaranteeing hamiltonicity of claw-free graphs (as reflected in Theorems 2.4 and 2.5 of this section). To formulate Chen's results, we need some additional notation.…”

“…As a special case of Theorem 2.4 with fixed given values for p and ε, the following was obtained in [34], and independently in [79]. Theorem 2.5.…”

“…It is known that 0 (5, 2) = {K 2,3 }, and that for k ≥ 4 these classes are empty, but for other appropriate values of k and r these classes are usually not easy to describe explicitly. In [34], Chen proved the following general result. [34]).…”

“…In [34], Chen proved the following general result. [34]). Let p > 0 be a given integer, let ε be a given real number, and let k ∈ {2, 3}.…”

“…Motivated by Dirac's Theorem and Ore's Theorem, the related concept of the minimum degree sum over all independent sets of t vertices of a graph was introduced (See, e.g., [10,35,49,50,52,58,65,85]), as well as the minimum cardinality of the neighborhood union over all independent sets of t vertices of a graph (See, e.g., [1,45,47,64]), and the minimum cardinality of the neighborhood union over all sets of t vertices of a graph (See, e.g., [36,46,48]). Apart from the above concepts, other variations involve the maximum degree of pairs of vertices with distance two (independent pairs that have a common neighbor; Fan's condition; see, e.g., [43]), the minimum degree sum of any pairs of adjacent vertices (joined by an edge; see, e.g., [14,34,79]), and the maximum degree of pairs of adjacent vertices (Lai's condition; see, e.g., [59]). There exist several survey papers on hamiltonian graph theory in which the interested reader can find more details on the above concepts and conditions (See, e.g., the surveys in [3, 5, 7, 11, 44, 53-55, 57, 62]).…”