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What is the exact difference between the adjacent strong edge coloring and vertex distinguishing proper edge coloring of graphs? This paper refers to the fact that the adjacent strong edge coloring of all graphs without $K_2$ or $C_5$ component is less than $\Delta+2$ where $\Delta$ is the maximum degree of the graph as an open conjecture, but, whereas this paper gives a proof that the vertex distinguishing proper edge coloring is less than $\Delta+2$. But, the definitions seem to coincide. Am I missing something here? Thanks beforehand.

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    $\begingroup$ Without reading anything very closely (sincerely) it would seem that the 1999 paper proved the number is at most $|V(G)| + 1$ whereas the 2002 paper strengthens this to an upper bound of $\Delta + 2$. No? $\endgroup$ – Pat Devlin May 27 at 13:58
  • $\begingroup$ @PatDevlin oh so silly! I was in the mood of complete graph($\Delta=n-1$), thereby implying that $|V(G)+1|=\Delta+2$. Anyways, so the two terms(vertex distinguishing and adjacent strong edge coloring) are synonymous right? $\endgroup$ – vidyarthi May 28 at 7:04

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