Geodesic
On integral sum graphs with a saturated vertex
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 669-674 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

As introduced by F. Harary in 1994, a graph $ G$ is said to be an $integral$ $ sum$ $ graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $ f(u)+f(v)=f(w)$ for some vertex $w$ in $G$. \endgraf We prove that every integral sum graph with a saturated vertex, except the complete graph $K_3$, has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be {\it saturated} if it is adjacent to every other vertex of $G$.) Some direct corollaries are also presented.
As introduced by F. Harary in 1994, a graph $ G$ is said to be an $integral$ $ sum$ $ graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $ f(u)+f(v)=f(w)$ for some vertex $w$ in $G$. \endgraf We prove that every integral sum graph with a saturated vertex, except the complete graph $K_3$, has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be {\it saturated} if it is adjacent to every other vertex of $G$.) Some direct corollaries are also presented.
Classification : 05C15, 05C78
Keywords: integral sum graph; saturated vertex; edge-chromatic number 
@article{CMJ_2010_60_3_a5,
     author = {Chen, Zhibo},
     title = {On integral sum graphs with a saturated vertex},
     journal = {Czechoslovak Mathematical Journal},
     pages = {669--674},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672408},
     zbl = {1224.05439},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a5/}
}
TY  - JOUR
AU  - Chen, Zhibo
TI  - On integral sum graphs with a saturated vertex
JO  - Czechoslovak Mathematical Journal
PY  - 2010
SP  - 669
EP  - 674
VL  - 60
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a5/
LA  - en
ID  - CMJ_2010_60_3_a5
ER  - 
%0 Journal Article
%A Chen, Zhibo
%T On integral sum graphs with a saturated vertex
%J Czechoslovak Mathematical Journal
%D 2010
%P 669-674
%V 60
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a5/
%G en
%F CMJ_2010_60_3_a5
Chen, Zhibo. On integral sum graphs with a saturated vertex. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 669-674. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a5/

[1] Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications. Macmillan, London (1976). | MR

[2] Chartrand, G., Lesniak, L.: Graphs and Digraphs, 2nd ed. Wadsworth., Belmont (1986). | MR | Zbl

[3] Chen, Z.: Integral sum graphs from identification. Discrete Math. 181 (1998), 77-90. | DOI | MR | Zbl

[4] Chen, Z.: On integral sum graphs. Discrete Math. 306 (2006), 19-25 (It first appeared in The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications, 11 pp. (electronic), Electron. Notes Discrete Math., 11, Elsevier, Amsterdam, 2002.). | MR | Zbl

[5] Ellingham, M. N.: Sum graphs from trees. Ars Combin. 35 (1993), 335-349. | MR | Zbl

[6] Gallian, J.: A dynamic survey of graph labeling. Electronic J. Combinatorics 15 (2008). | MR

[7] Harary, F.: Sum graphs and difference graphs. Congr. Numer. 72 (1990), 101-108. | MR | Zbl

[8] Harary, F.: Sum graphs over all the integers. Discrete Math. 124 (1994), 99-105. | DOI | MR | Zbl

[9] He, W., Wang, L., Mi, H., Shen, Y., Yu, X.: Integral sum graphs from a class of trees. Ars Combinatoria 70 (2004), 197-205. | MR | Zbl

[10] Imrich, W., Klavar, S.: Product Graphs. John Wiley & Sons, New York (2000). | MR

[11] Liaw, S.-C., Kuo, D., Chang, G.: Integral sum numbers of Graphs. Ars Combin. 54 (2000), 259-268. | MR | Zbl

[12] Mahmoodian, E. S.: On edge-colorability of Cartesian products of graphs. Canad. Math. Bull. 24 (1981), 107-108. | DOI | MR | Zbl

[13] Pyatkin, A. V.: Subdivided trees are integral sum graphs. Discrete Math. 308 (2008), 1749-1750. | DOI | MR | Zbl

[14] Sharary, A.: Integral sum graphs from complete graphs, cycles and wheels. Arab Gulf Sci. Res. 14-1 (1996), 1-14. | MR | Zbl

[15] Thomassen, C.: Hajós conjecture for line graphs. J. Combin. Theory Ser. B 97 (2007), 156-157. | DOI | MR | Zbl