Two graphs G and H are hypomorphic if there exists a bijection φ : V (G)→ V (H) such that G− v ∼= H − φ(v) for each v ∈ V (G). A graph G is reconstructible if H ∼= G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we con...

A graph is said to be a sum graph if there exists a set S of positive integers as its node set, with two nodes adjacent whenever their sum is in S. An integral sum graph is defined just as the sum graph, the difference being that S is a subset of 2~ instead of N*. The sum number of a given graph G is defined as the smallest number of isolated nodes which when added to G result in a sum graph. T...

In the early 60s, Harary and Hill conjectured H(n) := 1 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c to be the minimum number of crossings among all drawings of the complete graph Kn. It has recently been shown that this conjecture holds for so-called shellable drawings of Kn. For n ≥ 11 odd, we construct a non-shellable family of drawings of Kn with exactly H(n) crossings. In particular, every edge in our dr...

In this note, we show that one of the arguments used by Moazzami for computing the tenacity of the third Harary graph is wrong and then improve the proof. © 2014 Published by Elsevier B.V.

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the fo...