[Paper Review] On the circumference, connectivity and dominating cycles

Zh. G. Nikoghosyan|ArXiv.org|Jun 10, 2009

Advanced Graph Theory Research5 references3 citations

TL;DR

This paper establishes a reverse Dirac-type result for 4-connected graphs: every such graph either contains a cycle of length at least $4\delta - 2\kappa$ or has a dominating cycle. The proof relies on analyzing endfragments and their structural properties, extending prior work on circumference and connectivity, and confirms the tightness of the bound via extremal graph constructions.

ABSTRACT

Every 4-connected graph with minimum degree $δ$ and connectivity $κ$ either has a cycle of length at least $4δ-2κ$ or has a dominating cycle.

Motivation & Objective

• To establish a reverse version of the 4-connected Hamiltonian cycle theorem, focusing on circumference and dominating cycles.
• To investigate whether the bound $4\delta - 2\kappa$ for cycle length in 4-connected graphs is tight and necessary.
• To extend prior results on dominating cycles in graphs with higher connectivity and minimum degree constraints.
• To demonstrate the optimality of the bound through extremal graph constructions, including $4K_2 + K_3$, $5K_2 + K_4$, and $H(1,n-2\delta,\delta,\kappa)$.
• To unify and generalize earlier results on endfragments and fragment structures in 4-connected graphs.

Proposed method

• Analyzes endfragments $A^{\uparrow}$ and $A^{\downarrow}$ in 4-connected graphs, using their size constraints to derive cycle length bounds.
• Applies fragment theory via $N(X)$ and $\hat{X}$ to define minimum cut-sets and analyze structural components.
• Uses case analysis based on inclusion of fragments in vertex sets $V^{\uparrow}$ and $V^{\downarrow}$, with subcases based on independence and size conditions.
• Employs inequalities derived from degree and connectivity constraints to bound cycle length $c \geq 4\delta - 2\kappa$.
• Applies lemmas (e.g., Lemma 9, Lemma 10) to establish lower bounds on $|V^{\uparrow}|$ and $|V^{\downarrow}|$ under different fragment inclusion scenarios.
• Combines results from four generalized theorems (Theorems 2–5) to cover all structural cases, leading to the main result in Theorem 1.

Experimental results

Research questions

• RQ1 Can the bound $4\delta - 2\kappa$ for cycle length in 4-connected graphs be improved or tightened in the context of dominating cycles?
• RQ2 Is it possible to replace the 4-connectivity condition with 3-connectivity while preserving the cycle or dominating cycle conclusion?
• RQ3 Can the cycle length bound $4\delta - 2\kappa$ be strengthened to $4\delta - 2\kappa + 1$ without losing the guarantee of a long cycle or dominating cycle?
• RQ4 Is the conclusion that a dominating cycle exists optimal, or can it be replaced by a Hamilton cycle under the same conditions?
• RQ5 How do endfragment structures and their size constraints influence the existence of long cycles or dominating cycles in 4-connected graphs?

Key findings

• Every 4-connected graph either has a cycle of length at least $4\delta - 2\kappa$ or contains a dominating cycle.
• The 4-connectivity condition is tight: the graph $4K_2 + K_3$ is 3-connected but violates the conclusion, showing 4-connectivity is necessary.
• The bound $4\delta - 2\kappa$ is optimal: the graph $5K_2 + K_4$ shows that increasing the bound by 1 fails to guarantee a long cycle.
• The conclusion that a dominating cycle exists cannot be replaced by a Hamilton cycle: the graph $H(1,n-2\delta,\delta,\kappa)$ satisfies the conditions but lacks a Hamilton cycle.
• The proof relies on four generalized theorems (Theorems 2–5) that cover all possible configurations of endfragments in 4-connected graphs.
• The results confirm the reverse version of Theorem K (Sun, Tian, Wei) and support Conjecture 1 and Conjecture 2 on improved bounds for 4-connected graphs.