Maximum Degree Growth of the Iterated Line Graph
Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds. Niepel, Knor, and Šoltés have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices.
2009 ◽
Vol 85
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pp. 39-46
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2001 ◽
Vol 41
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pp. 1041-1045
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2019 ◽
Vol 19
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pp. 2050068
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2018 ◽
Vol 10
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pp. 1850069
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2020 ◽
Vol 584
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pp. 287-293
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2020 ◽
Vol 587
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pp. 291-301
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