The distinguishing index of the Cartesian product of finite graphs
Gorzkowska, Aleksandra ; Kalinowski, Rafał ; Pilśniak, Monika
Razlikovalni indeks ▫$D'(G)$▫ grafa ▫$G$▫ je najmanjše naravno število ▫$d$▫, pri katerem ima ▫$G$▫ barvanje povezav z ▫$d$▫ barvami, ki ga ohranja samo identični avtomorfizem. V tem članku ...obravnavamo razlikovalni indeks kartezičnega produkta povezanih končnih grafov. Dokažemo, da ima ▫$k$▫-ta kartezična potenca povezanega grafa ▫$G$▫ razlikovalni indeks enak 2 za vsak ▫$k \ge 2$▫, z edino izjemo ▫$D'(K_2^2) = 3$▫. Dokažemo tudi, da, če sta ▫$G$▫ in ▫$H$▫ povezana grafa, ki zadoščata relaciji ▫$2 \le |G| \le |H| \le 2^{|G|}(2^{|G|} - 1) - |G| + 2$▫, potem je ▫$D'(G \Box H) \le 2$▫ razen ▫$G \Box H = K_2^2$▫.
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