scholarly journals The super-connectivity of Kneser graphs

2019 ◽  
Vol 39 (1) ◽  
pp. 5 ◽  
Author(s):  
Gülnaz Boruzanli Ekinci ◽  
John Baptist Gauci
Keyword(s):  
Super Connectivity ◽  
Kneser Graphs

scholarly journals The super-connectivity of odd graphs and of their Kronecker double cover

2020 ◽  
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
John Baptist Gauci
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Kronecker Product ◽  
Double Cover ◽  
Kneser Graph ◽  
Classical Notion ◽  
Super Connectivity ◽  
Minimum Number ◽  
Kneser Graphs

The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super--connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size $k$ from the set of integers $\{1,\ldots,2k+1\}$ as vertices, and defining two vertices to be adjacent if the corresponding $k$-subsets are disjoint; these correspond to the Kneser graphs $KG(2k+1,k)$. The Kronecker double cover of a graph $G$ is formed by taking the Kronecker product of $G$ with the complete graph on two vertices; in the case when $G$ is $KG(2k+1,k)$, the Kronecker double cover is the bipartite Kneser graph $H(2k+1,k)$. We show that in both instances, the super--connectivity is equal to $2k$.

Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

2021 ◽  
Vol 344 (7) ◽  
pp. 112430
Author(s):  
Johann Bellmann ◽  
Bjarne Schülke
Keyword(s):  
Short Proof ◽  
Kneser Graphs

On the neighborhood complex of s→-stable Kneser graphs

2021 ◽  
Vol 344 (4) ◽  
pp. 112302
Author(s):  
Hamid Reza Daneshpajouh ◽  
József Osztényi
Keyword(s):  
Neighborhood Complex ◽  
Kneser Graphs

Kneser Graphs

2001 ◽  
pp. 135-161
Author(s):  
Chris Godsil ◽  
Gordon Royle
Keyword(s):  
Kneser Graphs

scholarly journals A Note on Hedetniemi's Conjecture, Stahl's Conjecture and the Poljak-Rödl Function

10.37236/8787 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Claude Tardif ◽  
Xuding Zhu
Keyword(s):  
Kneser Graphs ◽  
Hedetniemi’S Conjecture ◽  
Hedetniemi's Conjecture

We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$.

scholarly journals Super Connectivity of Erdős-Rényi Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 267 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graphs ◽  
Minimum Cardinality ◽  
Super Connectivity ◽  
Isolated Vertex ◽  
Disconnected Graph

The super connectivity κ ′ ( G ) of a graph G is the minimum cardinality of vertices, if any, whose deletion results in a disconnected graph that contains no isolated vertex. G is said to be r-super connected if κ ′ ( G ) ≥ r . In this note, we establish some asymptotic almost sure results on r-super connectedness for classical Erdős–Rényi random graphs as the number of nodes tends to infinity. The known results for r-connectedness are extended to r-super connectedness by pairing off vertices and estimating the probability of disconnecting the graph that one gets by identifying the two vertices of each pair.

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