Abstract
Given a proper edge-coloring of a loopless multigraph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. The palette index of a multigraph is defined as the minimum number of distinct palettes occurring among the vertices, taken over all proper edge-colorings of the multigraph itself. In this framework, the palette pseudograph of an edge-colored multigraph is defined in this paper and some of its properties are investigated. We show that these properties can be applied in a natural way in order to produce the first known family of multigraphs whose palette index is expressed in terms of the maximum degree by a quadratic polynomial. We also attempt an analysis of our result in connection with some related questions.
Highlights
Speaking, as soon as a chromatic parameter for graphs is introduced, the first piece of information that is retrieved is whether some universal meaningful upper or lower bound holds for it
To this purpose we introduce an additional tool, that we call the palette pseudograph, which can be defined from a given multigraph with a proper edge-coloring
The main purpose of this Section is the construction of a multigraph G∆ with maximum degree ∆, whose palette index is expressed by a quadratic polynomial in ∆
Summary
As soon as a chromatic parameter for graphs is introduced, the first piece of information that is retrieved is whether some universal meaningful upper or lower bound holds for it. In the current paper we make no exception to this trend and use the maximum degree ∆ as a reference value for the recently introduced chromatic parameter known as the palette index To this purpose we introduce an additional tool, that we call the palette pseudograph, which can be defined from a given multigraph with a proper edge-coloring. One cannot expect for the palette index any analogue of, say, Vizing’s theorem for the chromatic index: the palette index of graphs of maximum degree ∆ cannot admit a linear polynomial in ∆ as a universal upper bound It is the main purpose of the present paper to produce an infinite family of multigraphs, whose palette index grows asymptotically as ∆2, see Section 3. This concept is strictly related to the notion of palette index and it appears to yield a somewhat natural approach to the study of this chromatic parameter
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