Neighborhood Complexes of Some Exponential Graphs

Nilakantan, Nandini; Shukla, Samir

doi:10.37236/5312

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…In [2], Babson and Kozlov showed that neighborhood complex of complete graph K m is homotopy equivalent to the (m − 2)-sphere S m−2 . In [25], Nilakantan and the author have studied the neighborhood complexes of the exponential graphs K Kn m . We have shown that N (K Kn m ) is homotopy equivalent to S m−2 for m < n. For all the above mentioned classes of graphs the bound given in (2) is sharp.…”

Section: Neighborhood Complexmentioning

confidence: 99%

Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi

Shukla

2018

Preprint

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The neighborhood complex N (G) of a graph G were introduced by L. Lovász in his proof of Kneser conjecture. He proved that for any graph G, χ(G) ≥ conn(N (G)) + 3.(1)In this article we show that for a class of exponential graphs the bound given in (1) is tight. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs.Hedetniemi's conjecture states that the chromatic number of the categorical product of two graphs is the minimum of the chromatic number of the factors. Let M (G) denotes the Mycielskian of a graph G. We show that, for any graph G containing M (M (K n )) as a subgraph and for any graph H, if χ(G × H) = n + 1, then min{χ(G), χ(H)} = n + 1. Therefore, we enrich the family of graphs satisfying the Hedetniemi's conjecture.

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Section: Neighborhood Complexmentioning

confidence: 99%

Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi

Shukla

2018

Preprint

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“…In [3], Björner and Longueville showed that the neighborhood complexes of a family of vertex critical subgraphs of Kneser graphs -the stable Kneser graphs, are spheres up to homotopy. In [10], Nilakantan and author studied the neighborhood complexes of the exponential graphs K Kn n+1 . In this article we compute the homotopy type of the neighborhood complexes of 4-regular circulant graphs.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Type of the Neighborhood Complexes of Graphs of Maximal Degree at most $3$ and $4$-Regular Circulant Graphs

Shukla

2019

Electron. J. Combin.

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To estimate the lower bound for the chromatic number of a graph G, Lovász associated a simplicial complex N (G) called the neighborhood complex and relates the topological connectivity of N (G) to the chromatic number of G. More generally he proved that the chromatic number of G is bounded below by the topological connectivity of N (G) plus 3.In this article, we consider the graphs of maximal degree at most 3 and 4-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent either to a point, to a wedge sum of circles, to a wedge sum of 2-spheres S 2 , to S 3 , to a garland of 2-spheres S 2 or to a connected sum of tori.

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Neighborhood Complexes, Homotopy Test Graphs and an Application to Coloring of Product Graphs

Shukla

2022

Graphs and Combinatorics

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Neighborhood Complexes of Some Exponential Graphs

Abstract: In this article, we consider the bipartite graphs K 2 × K n . We first show that the connectedness of N (K Kn n+1 ) = 0. Further, we show that Hom(K 2 × K n , K m ) is homotopic to S m−2 , if 2 ≤ m < n.

Cited by 4 publications

References 9 publications

Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi

Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi

Homotopy Type of the Neighborhood Complexes of Graphs of Maximal Degree at most $3$ and $4$-Regular Circulant Graphs

Neighborhood Complexes, Homotopy Test Graphs and an Application to Coloring of Product Graphs

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