What question did this study set out to answer?

To establish the independence ratio for planar graphs with specific conditions related to cycles and triangles.

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March 17, 2026Open Access

The Independence Ratio of 4‐Cycle‐Free Planar Graphs

Key Points

To establish the independence ratio for planar graphs with specific conditions related to cycles and triangles.
Proving the independence ratio for planar graphs without triangles sharing edges with 4-cycles.
Extending results to 4-cycle-free and planar graphs with adjacent triangles and specific cycles.
The independence ratio for these graphs is established as a fixed value.
Strengthened bounds were provided for planar graphs with no adjacent triangles and 5-cycles.

Abstract

ABSTRACT We prove that every ‐vertex planar graph with no triangle sharing an edge with a 4‐cycle has independence ratio for . This result implies that the same bound holds for 4‐cycle‐free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5‐cycle. For the latter case we strengthen the bound to .

The Independence Ratio of 4‐Cycle‐Free Planar Graphs

Key Points

Abstract

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