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- W762451895 abstract "A simple graph $$G=(V,,E)$$ is said to be antimagic if there exists a bijection $$f{text {:}},Erightarrow [1,,|E|]$$ such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection $$f{text {:}},Vrightarrow [1,, |V|],$$ such that $$forall x,,yin V,$$ $$begin{aligned} sum _{x_iin N(x)}fleft( x_iright) ne sum _{x_jin N(y)}fleft( x_jright) . end{aligned}$$ Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval $$[1,,2n+m-4]$$ and, for trees with k inner vertices, in the interval $$[1,,m+k].$$ In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree $$Delta $$ in the interval $$[1,,n+t(n-t)],$$ where $$ t=min {Delta ,,lfloor n/2rfloor },$$ and, for trees with k leaves, in the interval $$[1,, 3n-4k].$$ In particular, all trees with $$n=2k$$ vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam." @default.
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- W762451895 date "2016-10-04" @default.
- W762451895 modified "2023-10-01" @default.
- W762451895 title "Approximate results for rainbow labelings" @default.
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- W762451895 doi "https://doi.org/10.1007/s10998-016-0151-2" @default.
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