Geodesic
The smallest one-realization of a given set
The electronic journal of combinatorics, Tome 19 (2012) no. 1
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For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we determine the number of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král in 2004.
DOI : 10.37236/1171
Classification : 05C15, 05C65, 05C85
Mots-clés : hypergraph coloring, mixed hypergraph, feasible set, chromatic spectrum, one-realization 
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     author = {Ping Zhao and Kefeng Diao and Kaishun Wang},
     title = {The smallest one-realization of a given set},
     journal = {The electronic journal of combinatorics},
     year = {2012},
     volume = {19},
     number = {1},
     doi = {10.37236/1171},
     zbl = {1243.05091},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1171/}
}
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Ping Zhao; Kefeng Diao; Kaishun Wang. The smallest one-realization of a given set. The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/1171

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