Total mixed domination in graphs

Total dominator total chromatic numbers of cycles and paths

“…Here, we calculate the total dominator total chromatic number of cycles. First we recall a proposition from [12] and calculate the mixed independence number of a cycle.…”

“…Because 2a 1 + a 2 ≥ 17, 10 ≤ a 1 + a 2 ≤ 12 and 5 ≤ a 1 ≤ 9 and max{0, 17 − 2a 1 , 10 − a 1 } ≤ a 2 ≤ min{7, 12 − a 1 } by Facts 5-8. Since (a 1 , a 2 ) = (5, 7), (6,6), (7,5), (8,4), (9, 3) imply 12 i=1 |V i | ̸ = 2n, and (a 1 , a 2 ) = (6, 5), (7,4), (8,3), (9, 1), (9, 2) imply |V 1 | > α = 11, which contradict…”

“…Since T (P 2 ) ∼ = K 3 and T (P 3 ) contains K 3 as a subgraph and ({v 1 , e 23 }, {v 3 , e 12 }, {v 2 }) is a TDC of T (P 3 ), we have χ tt d (P n ) = χ t d (T (P n )) = 3 for n = 2, 3. We know from [12] that for n ≥ 4 the sets…”

“…Since the vertices v 12 and e (11) (12) did not totally dominated by a color class of cardinality one, we must have CN (V 3 ) = {v 12 , e (11)(12) } which is not possible. Thus ℓ ≥ 10, which implies χ tt d (P 12 ) = 10 = γ tm (P 12 ) + 3 by (3.0.1).…”

“…A total dominating set, briefly TDS, S of a graph G = (V, E) is a subset of the vertex set of G such that for each vertex v, N G (v) ∩ S ̸ = ∅. The total domination number γ t (G) of G is the minimum cardinality of a TDS of G. Similarly, a subset S ⊆ V ∪ E of a graph G is called a total mixed dominating set, briefly TMDS, if each object of (V ∪ E) is either adjacent or incident to an object of S, and the total mixed domination number γ tm (G) of G is the minimum cardinality of a TMDS of G [12]. A min-TDS/ min-TMDS of G denotes a TDS/ TMDS of G with minimum cardinality.…”