Let G be a 4-connected graph, let Ẽ(G) denote the set of those edges of G which are not contained in a triangle, and let Ec(G) denote the set of 4-contractible edges of G. We show that if |Ẽ(G)| ≥ 1, then |Ec(G)| ≥ (|Ẽ(G)|+ 8)/4 unless G satisfies certain conditions.

A matroid M is said to be k–connected up to separators of size l if whenever A is (k − 1)–separating in M , then either |A| ≤ l or |E(M)− A| ≤ l. We use si(M) and co(M) to denote the simplification and cosimplification of the matroid M . We prove that if a 3–connected matroid M is 4–connected up to separators of size 5, then there is an element x of M such that either co(M\x) or si(M/x) is 3–co...

In this paper, we first present a new variation of hypercubes, denoted by SQn. SQn is obtained from Qn by changing some links. SQn is also an n-regular n-connected graph but of diameter about n/4. Then, we present a generalization of SQn. For any positive integer g, we can construct an n-dimensional generalized shuffle-cube with 2n vertices which is n-regular and n-connected. However its diamet...

Tutte’s Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| − |E(N)| = 1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4connected, that is, for every 3-separation (X, Y ) of N , the set E(N) can be ...

In this paper, we describe an O(|V (G)|) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed networks, as well as the study of non-separating induced paths in highly connected graphs. Supporte...

For a connected graph G of order n and minimum degree δ we prove the existence of two disjoint dominating sets D1 and D2 such that, if δ ≥ 2, then |D1∪D2| ≤ 67n unless G = C4, and, if δ ≥ 5, then |D1 ∪ D2| ≤ 2 δ+1 n. While for the first estimate there are exactly six extremal graphs which are all of order 7, the second estimate is asymptotically best-possible.

5 Archdeacon proved that projective-planar graphs are characterized by 35 excluded minors. 6 Using this result we show that internally 4-connected projective-planar graphs are characterized 7 by 23 internally 4-connected excluded minors. 8

The Randić index R(G) of a graph G is the sum of weights (deg(u) deg(v))−0.5 over all edges uv of G, where deg(v) denotes the degree of a vertex v. Let r(G) be the radius of G. We prove that for any connected graph G of maximum degree four which is not a path with even number of vertices, R(G) ≥ r(G). As a consequence, we resolve the conjecture R(G) ≥ r(G)− 1 given by Fajtlowicz in 1988 for the...

In an earlier paper, we proved that an internally 4connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from ...

The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if C is a hamiltonian cycle of a graph G, then the square of C is called a hamiltonian square of G. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square, in a 4-connected graph. The connectivi...