A 3-consecutive C-coloring of a graph G = (V, E) is a mapping ϕ : V → N such that every path on three vertices has at most two colors. We prove general estimates on the maximum numberχ 3CC (G) of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs withχ 3CC (G) ≥ k for k = 3 and k = 4.
The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].
Three edges e 1 , e 2 and e 3 in a graph G are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e 1 , e 2 and e 3 are consecutive edges in G, then c(e 1) c(e 3). The injective edge coloring number χ i (G) is the minimum number of colors permitted in such a coloring. In this paper, exact values of χ i (G) for several classes of graphs are obtained, upper and lower bounds for χ i (G) are introduced and it is proven that checking whether χ i (G) = k is NP-complete.
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