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Regularity and Planarity of Token Graphs

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Carballosa W., Fabila-Monroy R., Leaños J., Rivera L. M.
Discussiones Mathematicae Graph Theory
|
2017
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tom 37
|
nr 3
573-586
EN
Let G = (V, E) be a graph of order n and let 1 ≤ k < n be an integer. The k-token graph of G is the graph whose vertices are all the k-subsets of V, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper we characterize precisely, for each value of k, which graphs have a regular k-token graph and which connected graphs have a planar k-token graph.
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The competition numbers of Johnson graphs

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Kim S.-R., Park B., Sano Y.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 3
449-459
EN
The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and to characterize all graphs with given competition number k has been one of the important research problems in the study of competition graphs. The Johnson graph J(n,d) has the vertex set ${v_X | X ∈ \binom{[n]}{d}$, where $\binom{[n]}{d}$ denotes the set of all d-subsets of an n-set [n] = {1,..., n}, and two vertices $v_{X₁}$ and $v_{X₂}$ are adjacent if and only if |X₁ ∩ X₂| = d - 1. In this paper, we study the edge clique number and the competition number of J(n,d). Especially we give the exact competition numbers of J(n,2) and J(n,3).
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