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This work proves a graph property with equal degree endpoints connected by a path, indicating deeper structural insights.
We address a problem posed by Erdős and Hajnal in 1991, proving that for all n ≥ 600 , every ( 2 n + 1 ) -vertex graph with at least n 2 + n + 1 edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph K n , n + 1 demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.
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Chen et al. (2026) studied this question.
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