Degree-continuous graphs

Gimbel, John, and Zhang, Ping. "Degree-continuous graphs." Czechoslovak Mathematical Journal 51.1 (2001): 163-171. <http://eudml.org/doc/30623>.

@article{Gimbel2001,
abstract = {A graph $G$ is degree-continuous if the degrees of every two adjacent vertices of $G$ differ by at most 1. A finite nonempty set $S$ of integers is convex if $k \in S$ for every integer $k$ with $\min (S) \le k \le \max (S)$. It is shown that for all integers $r > 0$ and $s \ge 0$ and a convex set $S$ with $\min (S) = r$ and $\max (S) = r+s$, there exists a connected degree-continuous graph $G$ with the degree set $S$ and diameter $2s+2$. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph $G$ and convex set $S$ of positive integers containing the integer 2, there exists a connected degree-continuous graph $H$ with the degree set $S$ and containing $G$ as an induced subgraph if and only if $\max (S)\ge \Delta (G)$ and $G$ contains no $r-$regular component where $r = \max (S)$.},
author = {Gimbel, John, Zhang, Ping},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance; degree-continuous; distance; degree-continuous},
language = {eng},
number = {1},
pages = {163-171},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Degree-continuous graphs},
url = {http://eudml.org/doc/30623},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Gimbel, John
AU - Zhang, Ping
TI - Degree-continuous graphs
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 1
SP - 163
EP - 171
AB - A graph $G$ is degree-continuous if the degrees of every two adjacent vertices of $G$ differ by at most 1. A finite nonempty set $S$ of integers is convex if $k \in S$ for every integer $k$ with $\min (S) \le k \le \max (S)$. It is shown that for all integers $r > 0$ and $s \ge 0$ and a convex set $S$ with $\min (S) = r$ and $\max (S) = r+s$, there exists a connected degree-continuous graph $G$ with the degree set $S$ and diameter $2s+2$. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph $G$ and convex set $S$ of positive integers containing the integer 2, there exists a connected degree-continuous graph $H$ with the degree set $S$ and containing $G$ as an induced subgraph if and only if $\max (S)\ge \Delta (G)$ and $G$ contains no $r-$regular component where $r = \max (S)$.
LA - eng
KW - distance; degree-continuous; distance; degree-continuous
UR - http://eudml.org/doc/30623
ER -