# Difference labelling of digraphs

Discussiones Mathematicae Graph Theory (2004)

- Volume: 24, Issue: 3, page 509-527
- ISSN: 2083-5892

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topMartin Sonntag. "Difference labelling of digraphs." Discussiones Mathematicae Graph Theory 24.3 (2004): 509-527. <http://eudml.org/doc/270712>.

@article{MartinSonntag2004,

abstract = {A digraph G is a difference digraph iff there exists an S ⊂ N⁺ such that G is isomorphic to the digraph DD(S) = (V,A), where V = S and A = \{(i,j):i,j ∈ V ∧ i-j ∈ V\}.For some classes of digraphs, e.g. alternating trees, oriented cycles, tournaments etc., it is known, under which conditions these digraphs are difference digraphs (cf. [5]). We generalize the so-called source-join (a construction principle to obtain a new difference digraph from two given ones (cf. [5])) and construct a difference labelling for the source-join of an even number of difference digraphs. As an application we obtain a sufficient condition guaranteeing that certain (non-alternating) trees are difference digraphs.},

author = {Martin Sonntag},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {graph labelling; difference digraph; oriented tree; graph labeling},

language = {eng},

number = {3},

pages = {509-527},

title = {Difference labelling of digraphs},

url = {http://eudml.org/doc/270712},

volume = {24},

year = {2004},

}

TY - JOUR

AU - Martin Sonntag

TI - Difference labelling of digraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2004

VL - 24

IS - 3

SP - 509

EP - 527

AB - A digraph G is a difference digraph iff there exists an S ⊂ N⁺ such that G is isomorphic to the digraph DD(S) = (V,A), where V = S and A = {(i,j):i,j ∈ V ∧ i-j ∈ V}.For some classes of digraphs, e.g. alternating trees, oriented cycles, tournaments etc., it is known, under which conditions these digraphs are difference digraphs (cf. [5]). We generalize the so-called source-join (a construction principle to obtain a new difference digraph from two given ones (cf. [5])) and construct a difference labelling for the source-join of an even number of difference digraphs. As an application we obtain a sufficient condition guaranteeing that certain (non-alternating) trees are difference digraphs.

LA - eng

KW - graph labelling; difference digraph; oriented tree; graph labeling

UR - http://eudml.org/doc/270712

ER -

## References

top- [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. Zbl0702.05072
- [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. Zbl0785.05075
- [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. Zbl0574.05040
- [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. Zbl0484.05059
- [5] R.B. Eggleton and S.V. Gervacio, Some properties of difference graphs, Ars Combin. 19A (1985) 113-128. Zbl0562.05026
- [6] M.N. Ellingham, Sum graphs from trees, Ars Combin. 35 (1993) 335-349. Zbl0779.05042
- [7] S.V. Gervacio, Which wheels are proper autographs?, Sea Bull. Math. 7 (1983) 41-50. Zbl0524.05054
- [8] S.V. Gervacio, Difference graphs, in: Proc. of the Second Franco-Southeast Asian Math. Conf., Univ. of the Philippines, May 17-June 5, 1982.
- [9] 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-Intersci. Publ., Wiley, New York, 1991) 553-562. Zbl0840.05042
- [10] T. Hao, On sum graphs, J. Combin. Math. and Combin. Computing 6 (1989) 207-212. Zbl0701.05047
- [11] F. Harary, Sum graphs and difference graphs, Congress. Numer. 72 (1990) 101-108. Zbl0691.05038
- [12] F. Harary, Sum graphs over all the integers, Discrete Math. 124 (1994) 99-105, doi: 10.1016/0012-365X(92)00054-U. Zbl0797.05069
- [13] 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. Zbl0835.05075
- [14] 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. Zbl0791.05090
- [15] 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. Zbl0827.05048
- [16] M. Miller, J. Ryan and W.F. Smyth, The sum number of the cocktail party graph, Bull. Inst. Comb. Appl. 22 (1998) 79-90. Zbl0894.05048
- [17] 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. Zbl0918.05091
- [18] W.F. Smyth, Sum graphs of small sum number, Coll. Math. Soc. János Bolyai, 60. Sets, Graphs and Numbers, Budapest (1991) 669-678. Zbl0792.05120
- [19] W.F. Smyth, Sum graphs: new results, new problems, Bulletin of the ICA 2 (1991) 79-81. Zbl0828.05054
- [20] W.F. Smyth, Addendum to: ``Sum graphs: new results, new problems'', Bulletin of the ICA 3 (1991) 30. Zbl0828.05055
- [21] M. Sonntag, Difference labelling of cacti, Discuss. Math. Graph Theory 23 (2003) 55-65, doi: 10.7151/dmgt.1185. Zbl1054.05090
- [22] M. Sonntag, Difference labelling of the generalized source-join of digraphs, Preprint Series of TU Bergakademie Freiberg, Faculty of Mathematics and Computer Science, Preprint 2003-03 (2003) 1-18, ISSN 1433-9307.
- [23] M. Sonntag and H.-M. Teichert, Sum numbers of hypertrees, Discrete Math. 214 (2000) 285-290, doi: 10.1016/S0012-365X(99)00307-6. Zbl0943.05071
- [24] M. Sonntag and H.-M. Teichert, On the sum number and integral sum number of hypertrees and complete hypergraphs, Discrete Math. 236 (2001) 339-349, doi: 10.1016/S0012-365X(00)00452-0. Zbl0995.05106
- [25] H.-M. Teichert, The sum number of d-partite complete hypergraphs, Discuss. Math. Graph Theory 19 (1999) 79-91, doi: 10.7151/dmgt.1087. Zbl0933.05104
- [26] H.-M. Teichert, Classes of hypergraphs with sum number 1, Discuss. Math. Graph Theory 20 (2000) 93-104, doi: 10.7151/dmgt.1109.
- [27] H.-M. Teichert, Sum labellings of cycle hypergraphs, Discuss. Math. Graph Theory 20 (2000) 255-265, doi: 10.7151/dmgt.1124. Zbl0982.05070
- [28] H.-M. Teichert, Summenzahlen und Strukturuntersuchungen von Hypergraphen (Berichte aus der Mathematik, Shaker Verlag Aachen, 2001).

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