What question did this study set out to answer?

The aim is to prove the domination conjecture for 4-regular graphs that lack induced 4-cycles.

March 10, 2026

Domination in 4‐Regular Graphs With No Induced 4‐Cycles

Key Points

The aim is to prove the domination conjecture for 4-regular graphs that lack induced 4-cycles.
Defined a dominating set and the domination number in graph theory.
Analyzed structural properties of 4-regular graphs.
Applied combinatorial techniques to establish proof.
Successfully proved the conjecture in 4-regular graphs.
Demonstrated that the domination number meets the stated condition when no induced 4-cycles are present.

Abstract

ABSTRACT A set of vertices in a graph is a dominating set of if every vertex not in is adjacent to a vertex in . The domination number of , denoted by , is the minimum cardinality of a dominating set in . The ‐conjecture for domination in 4‐regular graphs states that if is a 4‐regular graph of order , then . We prove this conjecture when has no induced 4‐cycle.

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Domination in 4‐Regular Graphs With No Induced 4‐Cycles

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Abstract

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