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Extremal Bicyclic 3-Chromatic Graphs

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Abstract

A partial order relation in the set \(\mathcal {G}(n,k)\) of graphs of order \(n\) and chromatic number \(k\) can be defined as follows: Let \(G\) and \(H\) be two graphs in \(\mathcal {G}(n,k)\). \(G\) is said to be less than \(H\) if \(c_i(G)\le c_i(H)\) holds for every \(i\), \(k\le i\le n\) and at least one inequality is strict, where \(c_i(G)\) denotes the number of \(i\)-color partitions of \(G\). These numbers are the coefficients of the chromatic polynomial in factorial form. In (J Graph Theory 43:210–222, 2003) the first \(\lceil n/2\rceil \) levels of the diagram of the partially ordered set of connected 3-chromatic graphs of order \(n\) were described. In this paper the previous work is continued and a description of the \((\lceil n/2\rceil +1)\)-st level is given; it contains \(n/2+1\) bicyclic graphs for even \(n\) and \((n-1)/2\) bicyclic graphs for odd \(n\). Some consequences concerning ordering chromatic polynomials of these graphs are deduced.

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Correspondence to Sana Javed.

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Tomescu, I., Javed, S. Extremal Bicyclic 3-Chromatic Graphs. Graphs and Combinatorics 31, 1043–1052 (2015). https://doi.org/10.1007/s00373-014-1421-5

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  • DOI: https://doi.org/10.1007/s00373-014-1421-5

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