Christian RUBIO ( UNAM, Mexiko ):
 Geometric achromatic and pseudochromatic index
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Abstract
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Let $G$ be a simple graph. A (vertex) coloring of $G$ is proper if
no two adjacent vertices have the same color. Moreover if each pair
of colors appears on at least one pair of adjacent vertices, then the
coloring is complete. The biggest number $k$ for which a complete
coloring of (the vertices of) $G$ exist is the pseudoachromatic number
$\psi(G)$ of $G$. The biggest number $k$ for which a complete and
proper coloring of (the vertices of) $G$ exist is the achromatic number
$\alpha(G)$ of $G$. Clearly we have that
$$\chi(G) \leq \alpha(G) \leq \psi(G)$$ where $\chi(G)$ denotes, as
usual, the chromatic number of $G$.
The pseudoachromatic index $\psi_1(G)$ of $G$ is defined as the
pseudoachromatic number of the line graph $L(G)$ of $G$. The achromatic
index $\alpha_1(G)$ of $G$ is defined as the achromatic number of the
line graph of $G$. Notationally, $\psi_1(G) = \psi(L(G))$ and
$\alpha_1(G) = \alpha(L(G))$.
Bouchet in 1978 proved that if $q$ is an odd number and $n=q^2 + q + 1$,
the projective plane of order $q$ exist if and only if
$\alpha_1(K_n) = qn$. This result has motivated the study of the
achromatic and pseudoachromatic indices in complete graphs. In this
paper we extend the notion of pseudoachromatic and achromatic index to
geometric graphs and we define the geometric achromatic and
pseudoachromatic index for a graph $G$. Particularly, we present upper
and lower bounds for the achromatic and pseudoachromatic indices of
complete geometric graphs.