Algebraic characterization of uniquely vertex colorable graphs
نویسندگان
چکیده
منابع مشابه
Algebraic characterization of uniquely vertex colorable graphs
The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1 ∈ IG,k for a certain ideal IG,k ⊆ k[x1, . . . , xn]. In this paper, we extend this result by proving a general decomposition theorem for IG,k . This theorem allows us to giv...
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The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, k-colorability of a graph can be characterized in terms of whether its graph polynomial is contained in a certain ideal. In this paper, we interpret unique colorability in an analogous manner and prove an algebraic characterization for uniquely k-colo...
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This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...
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There is a conjecture due to Shaoji 3], about uniquely vertex r-colorable graphs which states: \ If G is a uniquely vertex r-colorable graph with order n and size (r ? 1)n ? ? r 2 , then G contains a K r as its subgraph." In this paper for any natural number r we construct a K r-free, uniquely r-colorable graph with (r ? 1)n ? ? r 2 edges. These families of graphs are indeed counter examples to...
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A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size χ(G) − 1. It is shown that...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2008
ISSN: 0095-8956
DOI: 10.1016/j.jctb.2007.08.004