# Graceful signed graphs: II. The case of signed cycles with connected negative sections

Mukti Acharya; Tarkeshwar Singh

Czechoslovak Mathematical Journal (2005)

- Volume: 55, Issue: 1, page 25-40
- ISSN: 0011-4642

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topAcharya, Mukti, and Singh, Tarkeshwar. "Graceful signed graphs: II. The case of signed cycles with connected negative sections." Czechoslovak Mathematical Journal 55.1 (2005): 25-40. <http://eudml.org/doc/30925>.

@article{Acharya2005,

abstract = {In our earlier paper [9], generalizing the well known notion of graceful graphs, a $(p,m,n)$-signed graph $S$ of order $p$, with $m$ positive edges and $n$ negative edges, is called graceful if there exists an injective function $f$ that assigns to its $p$ vertices integers $0,1,\dots ,q = m+n$ such that when to each edge $uv$ of $S$ one assigns the absolute difference $|f(u) - f(v)|$ the set of integers received by the positive edges of $S$ is $\lbrace 1,2,\dots ,m\rbrace $ and the set of integers received by the negative edges of $S$ is $\lbrace 1,2,\dots ,n\rbrace $. Considering the conjecture therein that all signed cycles $Z_k$, of admissible length $ k \ge 3$ and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths $ 0,2$ or $3\hspace\{4.44443pt\}(mod \; 4)$ in which the set of negative edges forms a connected subsigraph.},

author = {Acharya, Mukti, Singh, Tarkeshwar},

journal = {Czechoslovak Mathematical Journal},

keywords = {graceful signed graphs; signed cycles; graceful signed graphs; signed cycles},

language = {eng},

number = {1},

pages = {25-40},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Graceful signed graphs: II. The case of signed cycles with connected negative sections},

url = {http://eudml.org/doc/30925},

volume = {55},

year = {2005},

}

TY - JOUR

AU - Acharya, Mukti

AU - Singh, Tarkeshwar

TI - Graceful signed graphs: II. The case of signed cycles with connected negative sections

JO - Czechoslovak Mathematical Journal

PY - 2005

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 55

IS - 1

SP - 25

EP - 40

AB - In our earlier paper [9], generalizing the well known notion of graceful graphs, a $(p,m,n)$-signed graph $S$ of order $p$, with $m$ positive edges and $n$ negative edges, is called graceful if there exists an injective function $f$ that assigns to its $p$ vertices integers $0,1,\dots ,q = m+n$ such that when to each edge $uv$ of $S$ one assigns the absolute difference $|f(u) - f(v)|$ the set of integers received by the positive edges of $S$ is $\lbrace 1,2,\dots ,m\rbrace $ and the set of integers received by the negative edges of $S$ is $\lbrace 1,2,\dots ,n\rbrace $. Considering the conjecture therein that all signed cycles $Z_k$, of admissible length $ k \ge 3$ and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths $ 0,2$ or $3\hspace{4.44443pt}(mod \; 4)$ in which the set of negative edges forms a connected subsigraph.

LA - eng

KW - graceful signed graphs; signed cycles; graceful signed graphs; signed cycles

UR - http://eudml.org/doc/30925

ER -

## References

top- 10.1002/bs.3830030102, Behav. Sci. 3 (1958), 1–13. (1958) DOI10.1002/bs.3830030102
- Spectral criterion for cycle balance in networks, J. Graph Theory 4 (1981), 1–11. (1981) MR0558448
- 10.14429/dsj.32.6301, Def. Sci. J. 32 (1982), 231–236. (1982) Zbl0516.05055DOI10.14429/dsj.32.6301
- Are all polyominoes arbitrarily graceful? In: Graph Theory Singapore 1983, Lecture note in Mathematics, No. 1073, K. M. Koh, Y. P. Yap (eds.), Springer-Verlag, Berlin, 1984, pp. 205–211. (1984) MR0761019
- New algebraic models of social systems, Indian J. Pure Appl. Math. 17 (1986), 150–168. (1986) MR0830552
- 10.1002/jgt.3190140302, J. Graph Theory 14 (1990), 275–299. (1990) MR1060857DOI10.1002/jgt.3190140302
- On certain vertex valuations of a graph, Indian J. Pure Appl. Math. 22 (1991), 553–560. (1991) MR1124027
- $(k,d)$-graceful packings of a graph, In: Proc. of Group Discussion on graph labelling problems, Karnataka Regional Engineering College, Surathkal, August 16–25, 1999, B. D. Acharya, S M. Hegde (eds.).
- 10.1023/B:CMAJ.0000042369.18091.15, Czechoslovak Math. J. 54(129) (2004), 291–302. (2004) MR2059251DOI10.1023/B:CMAJ.0000042369.18091.15
- Line coloring of signed graphs, Elem. Math. 24 (1969), 49–52. (1969) MR0244098
- On a combinatorial problem of antennas in radio astronomy, In: Combinatorics; Proc. of the Colloquium of the Janos Bolyayi Mathematical Society (Keszthly; Hungary: 1976), Vol. 18, North-Holland, Amsterdam, 1978, pp. 135–149. (1978)
- Graphs as Mathematical Models, Prindle, Weber and Schmidt, Boston, Masschusetts, 1977. (1977) Zbl0384.05029MR0490611
- Application of Graph Theory to Group structures, Prentice Hall, Englewood Cliffs, 1963. (1963) MR0157785
- A dynamic survey of graph labelling, Electronic J. Comb., Dynamic Survey 8 (2001, DS6), 1–55. (2001, DS6)
- How to number a graph? In: Graph Theory and Computing, R. C. Read (ed.), Academic Press, New York, 1972, pp. 23–37. (1972) MR0340107
- On the notion of balance of a signed graph, Mich. Math. J. 2 (1954), 143–146. (1954) Zbl0056.42103MR0067468
- Structural Models: An Introduction to the Theory of Directed graphs, Wiley, New York, 1965. (1965) MR0184874
- Graph Theory, Addison-Wesley Publ. Comp., Reading Massachusetts, 1969. (1969) Zbl0196.27202MR0256911
- On certain vertex valuations of finite graphs, Utilitas Math. 4 (1973), 261–290. (1973) Zbl0277.05102MR0384616
- Graphs Associated with $[0,1]$ and $[0,+1,-1]$ Matrices, Department of Mathematics, Indian Institute of Technology, Bombay, 1974. (1974)
- Graph Theory and its Application to Problems of Society, SIAM, Philadelphia, 1978. (1978) MR0508050
- On certain valuations of the vertices of a graph, In: Theory of Graphs. Proc. Internat, Symp. (Rome, 1966), P. Rosentiehl (ed.), Dunod, Paris, 1968, pp. 349–355. (1968) MR0223271
- 10.1016/0012-365X(81)90066-2, Discrete Math. 34 (1981), 185–193. (1981) Zbl0461.05053MR0611431DOI10.1016/0012-365X(81)90066-2
- On $k$-graceful graphs, Congr. Numer. 36 (1982), 53–57. (1982) Zbl0519.05057MR0726049
- 10.1002/jgt.3190040202, J. Graph Theory 4 (1980), 127–144. (1980) Zbl0434.05059MR0570348DOI10.1002/jgt.3190040202
- 10.1016/0166-218X(82)90033-6, Discrete Appl. Math. 4 (1982), 47–74. (1982) Zbl0498.05030MR0676405DOI10.1016/0166-218X(82)90033-6
- A mathematical bibliography of signed and gain graphs and allied areas (manuscript prepared with Marge Pratt), Electronic J. Combinatorics 8 (1998). (1998) MR1744869

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