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## Discussiones Mathematicae Graph Theory

2003 | 23 | 1 | 55-65
Tytuł artykułu

### Difference labelling of cacti

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A graph G is a difference graph iff there exists S ⊂ IN⁺ such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = {{i,j}:i,j ∈ V ∧ |i-j| ∈ V}.
It is known that trees, cycles, complete graphs, the complete bipartite graphs $K_{n,n}$ and $K_{n,n-1}$, pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
55-65
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
• Faculty of Mathematics and Computer Science, TU Bergakademie Freiberg, Agricola-Str. 1, D-09596 Freiberg, Germany
Bibliografia
• [1] D. Bergstrand, F. Harary, K. Hodges, G. Jennings, L. Kuklinski and J. Wiener, The sum number of a complete graph, Bull. Malaysian Math. Soc. (Second Series) 12 (1989) 25-28.
• [2] D. Bergstrand, F. Harary, K. Hodges, G. Jennings, L. Kuklinski and J. Wiener, Product graphs are sum graphs, Math. Mag. 65 (1992) 262-264, doi: 10.2307/2691455.
• [3] G.S. Bloom and S.A. Burr, On autographs which are complements of graphs of low degree, Caribbean J. Math. 3 (1984) 17-28.
• [4] G.S. Bloom, P. Hell and H. Taylor, Collecting autographs: n-node graphs that have n-integer signatures, Annals N.Y. Acad. Sci. 319 (1979) 93-102, doi: 10.1111/j.1749-6632.1979.tb32778.x.
• [5] R.B. Eggleton and S.V. Gervacio, Some properties of difference graphs, Ars Combin. 19 (A) (1985) 113-128.
• [6] M.N. Ellingham, Sum graphs from trees, Ars Combin. 35 (1993) 335-349.
• [7] S.V. Gervacio, Which wheels are proper autographs?, Sea Bull. Math. 7 (1983) 41-50.
• [8] R.J. Gould and V. Rödl, Bounds on the number of isolated vertices in sum graphs, in: Y. Alavi, G. Chartrand, O.R. Ollermann and A.J. Schwenk, ed., Graph Theory, Combinatorics, and Applications 1 (Wiley, New York, 1991), 553-562.
• [9] T. Hao, On sum graphs, J. Combin. Math. and Combin. Computing 6 (1989) 207-212.
• [10] F. Harary, Sum graphs and difference graphs, Congressus Numerantium 72 (1990) 101-108.
• [11] F. Harary, Sum graphs over all the integers, Discrete Math. 124 (1994) 99-105, doi: 10.1016/0012-365X(92)00054-U.
• [12] F. Harary, I.R. Hentzel and D.P. Jacobs, Digitizing sum graphs over the reals, Caribb. J. Math. Comput. Sci. 1, 1 & 2 (1991) 1-4.
• [13] N. Hartsfield and W.F. Smyth, The sum number of complete bipartite graphs, in: R. Rees, ed., Graphs and Matrices (Marcel Dekker, New York, 1992), 205-211.
• [14] N. Hartsfield and W.F. Smyth, A family of sparse graphs of large sum number, Discrete Math. 141 (1995) 163-171, doi: 10.1016/0012-365X(93)E0196-B.
• [15] M. Miller, J. Ryan and W.F. Smyth, The sum number of the cocktail party graph, Bull. Inst. Comb. Appl. 22 (1998) 79-90.
• [16] M. Miller, Slamin, J. Ryan and W.F. Smyth, Labelling wheels for minimum sum number, J. Combin. Math. and Combin. Comput. 28 (1998) 289-297.
• [17] W.F. Smyth, Sum graphs of small sum number, Coll. Math. Soc. János Bolyai, 60. (Sets, Graphs and Numbers, Budapest, 1991) 669-678.
• [18] W.F. Smyth, Sum graphs: new results, new problems, Bulletin of the ICA 2 (1991) 79-81.
• [19] W.F. Smyth, Addendum to: 'Sum graphs: new results, new problems', Bulletin of the ICA 3 (1991) 30.
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