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

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Abstract

The neighborhood complex \(\mathcal {N}(G)\) of a graph G was introduced by L. Lovász in his proof of Kneser conjecture. He proved that for any graph G,

$$\begin{aligned} \chi (G) \ge conn(\mathcal {N}(G))+3. \end{aligned}$$
(2)

In this article we show that for a class of exponential graphs the bound given in (2) 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. In 1966, Hedetniemi conjectured that the chromatic number of the categori-cal product of two graphs is the minimum of the chromatic number of the factors. In 2019, Shitov [26] gave a counterexample to this conjecture. 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 \(\chi (G \times H) = n+1\), then \(\min \{\chi (G), \chi (H)\} = n+1\). Therefore, we enrich the family of graphs satisfying the Hedetniemi’s conjecture.

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Notes

  1. The generalized Mycielskians (also known as cones over graphs) of a graph were introduced by Tardif ([27]), which are the natural generalization of Mycielskian.

  2. The Mycielskian of a graph was introduced by Mycielski ([22]) in 1955.

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  1. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, India

    Samir Shukla

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Shukla, S. Neighborhood Complexes, Homotopy Test Graphs and an Application to Coloring of Product Graphs. Graphs and Combinatorics 38, 93 (2022). https://doi.org/10.1007/s00373-022-02490-2

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