A graph $G$ is uniquely $K_r$-saturated if it contains no clique with $r$ vertices and if for all edges $e$ in the complement, $G+e$ has a unique clique with $r$ vertices. Previously, few examples of uniquely $K_r$-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs. We find several new uniquely $K_r$-saturated graphs with $4 \leq r \leq 7$, as well as two new infinite families based on Cayley graphs for $\mathbb{Z}_n$ with a small number of generators.
@article{10_37236_2162,
author = {Stephen G. Hartke and Derrick Stolee},
title = {Uniquely {\(K_r\)-saturated} graphs},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {4},
doi = {10.37236/2162},
zbl = {1266.05057},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2162/}
}
TY - JOUR
AU - Stephen G. Hartke
AU - Derrick Stolee
TI - Uniquely \(K_r\)-saturated graphs
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2162/
DO - 10.37236/2162
ID - 10_37236_2162
ER -
%0 Journal Article
%A Stephen G. Hartke
%A Derrick Stolee
%T Uniquely \(K_r\)-saturated graphs
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2162/
%R 10.37236/2162
%F 10_37236_2162
Stephen G. Hartke; Derrick Stolee. Uniquely \(K_r\)-saturated graphs. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2162
