Minimum order of graphs with given coloring parameters

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $φ:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number $χ(G)$ and achromatic number $ψ(G)$, respectively), and the Grundy number $Γ(G)$ defined as the largest $k$ admitting a complete coloring $φ$ with exactly $k$ colors such that every vertex $v\in V$ of color $φ(v)$ has a neighbor of color $i$ for all $1\le i<φ(v)$. The inequality chain $χ(G)\le Γ(G)\le ψ(G)$ obviously holds for all graphs $G$. A triple $(f,g,h)$ of positive integers at least 2 is called realizable if there exists a connected graph $G$ with $χ(G)=f$, $Γ(G)=g$, and $ψ(G)=h$. Chartrand et al. (A note on graphs with prescribed complete coloring numbers, J. Combin. Math. Combin. Comput. LXXIII (2010) 77-84) found the list of realizable triples. In this paper we determine the minimum number of vertices in a connected graph with chromatic number $f$, Grundy number $g$, and achromatic number $h$, for all realizable triples $(f,g,h)$ of integers. Furthermore, for $f=g=3$ we describe the (two) extremal graphs for each $h \geq 6$. For $h=4$ and $5$, there are more extremal graphs, their description is contained as well.

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