Geodesic
$F$-continuous graphs
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 351-361
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph $G$ is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous graphs are determined. It is observed that if $G$ is a nontrivial connected graph that is $F$-continuous for all nontrivial connected graphs $F$, then either $G$ is regular or $G$ is a path. In the case of a 2-connected graph $F$, however, there always exists a regular graph that is not $F$-continuous. It is also shown that for every graph $H$ and every 2-connected graph $F$, there exists an $F$-continuous graph $G$ containing $H$ as an induced subgraph.
For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph $G$ is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous graphs are determined. It is observed that if $G$ is a nontrivial connected graph that is $F$-continuous for all nontrivial connected graphs $F$, then either $G$ is regular or $G$ is a path. In the case of a 2-connected graph $F$, however, there always exists a regular graph that is not $F$-continuous. It is also shown that for every graph $H$ and every 2-connected graph $F$, there exists an $F$-continuous graph $G$ containing $H$ as an induced subgraph.
Classification : 05C12, 05C38, 05C40, 05C75
Keywords: $F$-degree; $F$-degree continuous 
@article{CMJ_2001_51_2_a9,
     author = {Chartrand, Gary and Jarrett, Elzbieta B. and Saba, Farrokh and Salehi, Ebrahim and Zhang, Ping},
     title = {$F$-continuous graphs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {351--361},
     year = {2001},
     volume = {51},
     number = {2},
     mrnumber = {1844315},
     zbl = {0977.05042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a9/}
}
TY  - JOUR
AU  - Chartrand, Gary
AU  - Jarrett, Elzbieta B.
AU  - Saba, Farrokh
AU  - Salehi, Ebrahim
AU  - Zhang, Ping
TI  - $F$-continuous graphs
JO  - Czechoslovak Mathematical Journal
PY  - 2001
SP  - 351
EP  - 361
VL  - 51
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a9/
LA  - en
ID  - CMJ_2001_51_2_a9
ER  - 
%0 Journal Article
%A Chartrand, Gary
%A Jarrett, Elzbieta B.
%A Saba, Farrokh
%A Salehi, Ebrahim
%A Zhang, Ping
%T $F$-continuous graphs
%J Czechoslovak Mathematical Journal
%D 2001
%P 351-361
%V 51
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a9/
%G en
%F CMJ_2001_51_2_a9
Chartrand, Gary; Jarrett, Elzbieta B.; Saba, Farrokh; Salehi, Ebrahim; Zhang, Ping. $F$-continuous graphs. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 351-361. http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a9/

[1] G. Chartrand, L. Eroh, M. Schultz and P. Zhang: An introduction to analytic graph theory. Utilitas Math (to appear). | MR

[2] G. Chartrand, K. S.  Holbert, O. R.  Oellermann and H. C.  Swart: $F$-degrees in graphs. Ars Combin. 24 (1987), 133–148. | MR

[3] G.  Chartrand and L.  Lesniak: Graphs $\&$ Digraphs (third edition). Chapman $\&$ Hall, New York, 1996. | MR

[4] P.  Erdös and H.  Sachs: Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl. Wiss Z. Univ. Halle, Math-Nat. 12 (1963), 251–258. | MR

[5] J.  Gimbel and P.  Zhang: Degree-continuous graphs. Czechoslovak Math. J (to appear). | MR

[6] D.  König: Über Graphen und ihre Anwendung auf Determinantheorie und Mengenlehre. Math. Ann. 77 (1916), 453–465. | DOI | MR