Abstract
Hadwiger’s conjecture states that for every graph G, \(\chi (G)\le \eta (G)\), where \(\chi (G)\) is the chromatic number and \(\eta (G)\) is the size of the largest clique minor in G. In this work, we show that to prove Hadwiger’s conjecture in general, it is sufficient to prove Hadwiger’s conjecture for the class of graphs \(\mathcal {F}\) defined as follows: \(\mathcal {F}\) is the set of all graphs that can be expressed as the square graph of a split graph. Since split graphs are a subclass of chordal graphs, it is interesting to study Hadwiger’s Conjecture in the square graphs of subclasses of chordal graphs. Here, we study a simple subclass of chordal graphs, namely 2-trees and prove Hadwiger’s Conjecture for the squares of the same. In fact, we show the following stronger result: If G is the square of a 2-tree, then G has a clique minor of size \(\chi (G)\), where each branch set is a path.
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L. Sunil Chandran—Part of the work was done when this author was visiting Max Planck Institute for Informatics, Saarbruecken, Germany supported by Alexander von Humboldt Fellowship.
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Notes
- 1.
See Subsect. 1.3 for the definition.
- 2.
This generalization is in the same spirit as the generalization of graphs of maximum degree k to k-degenerate graphs. A graph G is a maximum degree k graph, if every vertex has at most k neighbors. A graph G is a k-degenerate graph is for any subset \(S \subseteq V(G)\), there exists a vertex \(u \in S\), such that u has at most k neighbors in G[S]. Graph classes which can be considered to be generalizations of quasi-line graphs can also be found in [KT14], for e.g. k-perfectly groupable graphs, k-simplicial graphs, k-perfectly orientable graphs etc.
- 3.
We omit proofs of some lemmas here due to space constraint. They can be found in the full version of the paper at http://arxiv.org/abs/1603.03205.
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Sunil Chandran, L., Issac, D., Zhou, S. (2016). Hadwiger’s Conjecture and Squares of Chordal Graphs. In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_34
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