Abstract
In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V,E) is P k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., colors) V 1 and V 2 such that for every induced P k (i.e., path with exactly k − 1 edges and k vertices) in G, the two colors alternate along the P k , i.e., no two consecutive vertices of the P k belong to the same color V i , i = 1,2. Obviously, a graph is bipartite if and only if is P 2-bicolorable, every graph is P k -bicolorable for some k and if G is P k -bicolorable then it is P k + 1-bicolorable. The notion of P k -bicolorable graphs is motivated by a similar notion of cycle-bicolorable graphs introduced in connection with chordal probe graphs. Moreover, P 3- and P 4-bicolorable graphs are closely related to various other concepts such as 2-subcolorable graphs, P 4-bipartite graphs and alternately orientable graphs.
We give a structural characterization of P 3-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P 4-bicolorable graphs in terms of forbidden subgraphs.
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Brandstädt, A., Golumbic, M.C., Le, V.B., Lipshteyn, M. (2009). Path-Bicolorable Graphs. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds) Graph Theory, Computational Intelligence and Thought. Lecture Notes in Computer Science, vol 5420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02029-2_17
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DOI: https://doi.org/10.1007/978-3-642-02029-2_17
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