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K-path irregular graphs. (English) Zbl 0669.05046

Combinatorics, graph theory, and computing, Proc. 19th Southeast. Conf., Boca Raton/Fla. 1988, Congr. Numerantium 65, 201-210 (1988).
[For the entire collection see Zbl 0665.00002.]
A connected graph \(G\) is \(k\)-path irregular, \(k\geq 1\), if every two vertices of \(G\) that are connected by a path of length \(k\) have distinct degrees. This extends the concepts of highly irregular (or 2-path irregular) graphs and totally segregated (or 1-path irregular) graphs. Various sets \(S\) of positive integers are considered for which there exist \(k\)-path irregular graphs for every \(k\in S\). It is shown for every graph \(G\) and every odd positive integer \(k\) that \(G\) can be embedded as an induced subgraph in a \(k\)-path irregular graph. Some open problems are also stated.

MSC:

05C38 Paths and cycles
05C99 Graph theory

Citations:

Zbl 0665.00002