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- W1568839932 abstract "Every $n$-vertex graph has two vertices with the same degree (if $nge2$). In general, let rep$(G)$ be the maximum multiplicity of a vertex degree in $G$. An easy counting argument yields rep$(G)ge n/(2d-2s+1)$, where $d$ is the average degree and $s$ is the minimum degree of $G$. Equality can hold when $2d$ is an integer, and the bound is approximately sharp in general, even when $G$ is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph. Among large claw-free graphs, repetition number $2$ is achievable, but if $G$ is an $n$-vertex line graph, then rep$(G)ge{1over4}n^{1/3}$. Among line graphs of trees, the minimum repetition number is $Theta(n^{1/2})$. For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear." @default.
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- W1568839932 date "2009-01-07" @default.
- W1568839932 modified "2023-09-25" @default.
- W1568839932 title "Repetition Number of Graphs" @default.
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- W1568839932 doi "https://doi.org/10.37236/96" @default.
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