×

The irregularity cost of a graph. (English) Zbl 0901.05086

A multigraph is called irregular if no two of its nodes have the same degree. If a graph \(G\) has at most one trivial component and no component isomorphic to \(K_2\), then there exists a multigraph \(H\) having \(G\) as underlying graph. We call such a multigraph \(H\) an irregular \(G\)-multigraph. For such a graph \(G\), the irregularity cost \(\text{ic}(G)\) is the minimum number of additional edges in an irregular \(G\)-multigraph. As an example, \(\text{ic} (K_{1,r})={r\choose 2}\). Theorem 1. Let \(G\) be a regular graph of order \(n\geq 4\) that contains a path of \(n-1\) nodes. Then \(\text{ic}(G)=(n^2-n)/4\) if \(n\equiv 0,1 \pmod 4\), \((n^2-n+2)/4\) if \(n\equiv 2,3 \pmod 4\). Theorem 2. For \(n\geq 4\) the irregularity cost of the path of order \(n\) is \(\text{ic}(P_n)= (n^2-3n+4)/4\) if \(n\equiv 0,3 \pmod 4\), \((n^2-3n+6)/4\) if \(n\equiv 1,2 \pmod 4\).

MSC:

05C99 Graph theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Behzad, M.; Chartrand, G., No graph is perfect, Amer. Math. Monthly, 74, 962-963 (1967) · Zbl 0179.52701
[2] Chartrand, G.; Jacobson, M. S.; Lehel, J.; Oellermann, O. R.; Ruiz, S.; Saba, F., Irregular networks, Congressus Numerantium, 64, 197-210 (1988)
[3] Jacobson, M. S.; Kubicka, E.; Kubicki, G., Irregularity sums for graphs, Vishwa Internat. J. Graph Theory, 1, 159-175 (1992)
[4] Harary, F.; Jacobson, M. S.; Kubicka, E.; Kubicki, G.; Oellermann, O. R., The irregularity cost or sum of a graph, Appl. Math. Lett., 6, 3, 79-80 (1993) · Zbl 0780.05052
[5] J. Lehel, Facts and quests on degree irregular assignments, Graph Theory, Combinatorics and Applications, (Edited by Y. Alavi et al.), Vol. 2, pp. 765-781, (1991).; J. Lehel, Facts and quests on degree irregular assignments, Graph Theory, Combinatorics and Applications, (Edited by Y. Alavi et al.), Vol. 2, pp. 765-781, (1991). · Zbl 0841.05052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.