Discrete Applied Mathematics· 2026Q2
A disproof of a gap-one conjecture for the equitable chromatic number of block graphs
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- 2026year
Short summary
A new construction disproves a conjecture that the equitable chromatic number of block graphs is at most one greater than a specific lower bound (L(G)).
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Key points
- The conjecture by Dybizbański et al. (2024) proposed L(G) ≤ χ=(G) ≤ L(G)+1 for block graphs.
- This paper constructs a connected block graph G_{d,k} for d ≥ 2 and k ≥ 4d-1.
- For these graphs, L(G_{d,k}) = k and χ=(G_{d,k}) = k+d.
- This shows the difference χ=(G) - L(G) is unbounded for connected block graphs.
AI-generated from the title and abstract; the full text is not read.
Abstract
For a graph G , let L ( G ) = max ω ( G ) , | V ( G ) | + 1 α min ( G ) + 1 , where ω ( G ) denotes the clique number, and α min ( G ) = min v ∈ V ( G ) max { | I | ∣ I is independent and v ∈ I } . Dybizbański, Furmańczyk, and Mkrtchyan (2024) conjectured that every block graph G satisfies L ( G ) ≤ χ = ( G ) ≤ L ( G ) + 1 , where χ = ( G ) is the equitable chromatic number of G . We disprove this conjecture in a strong form. For every pair of integers d ≥ 2 and k ≥ 4 d − 1 , we construct a connected block graph G d , k such that L ( G d , k ) = k and χ = ( G d , k ) = k + d . Thus the difference χ = ( G ) − L ( G ) is unbounded on connected block graphs.
The authors' abstract, as published at the source. Discrete Applied Mathematics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics