On the chromatic index of outerplanar graphs

The (2,1)-Total Labeling Number of Outerplanar Graphs Is at Most Δ + 2

“…(1.2) Assume that u ′ is a 3-vertex and v ′ is a 2-vertex (the other case can be treated similarly). Let f 1 : 5 be a (2,1)-total labeling obtained by applying Algorithm LABEL-K2 to G 3 as x 1 := u ′ ; f 1 (u ′ ) = 0. Moreover, as observed in (Case-1.1), we can obtain such a f 1 that f 1…”

“…Let u 2 be the neighbor of u 1 in G and u 3 , u 4 be vertices adjacent to u 2 in G −u 1 where u 3 = u 4 may occur (note that ∆(G) = 3). Hence we can extend f to the edge (u 1 , u 2 ) and the vertex u 1 as follows: assign a label a ∈ L 5…”

“…Let G 1 be a 2-connected component in G which has exactly one cut vertex v c of G. By δ(G) = 2, we have |V(G 1 )| ≥ 3 and hence d G 1 (v c ) ≥ 2. By κ(G) = 1 and ∆(G) = 3, it follows that d G 1 (v c ) = 2 holds and V(G 1 ) and V − V(G 1 ) are connected by a bridge incident to v c , where a bridge is an edge whose deletion makes the graph disconnected; denote this bridge by 5 . By symmetry, it suffices to consider the following three possible cases:…”

Algorithmic aspects of distance constrained labeling: a survey

“…Also, Bazzaro et al [5] showed that λ T p,1 (G) ≤ ∆+p+s for any sdegenerated graph (by χ(G) ≤ s+1 and χ ′ (G) ≤ ∆+1 [62]), where an s-degenerated graph G is a graph which can be reduced to a trivial graph by successive removal of vertices with degree at most s, that λ T p,1 (G) ≤ ∆ + p + 3 for any planar graph (by the Four-Color Theorem), and that λ T p,1 (G) ≤ ∆ + p + 1 for any outerplanar graph other than an odd cycle (since any outerplanar graph is 2-degenerated, and any outerplanar graph other than an odd cycle satisfies χ ′ (G) = ∆ [28]). As for the (2, 1)-total labeling number of outerplanar graphs is known to be at most ∆ + 2, which is tight, i.e., there exists an outerplanar graph whose (2, 1)-total labeling number is ∆ + 2 [36,37].…”

Parallel O(log n) time edge-colouring of trees and Halin graphs

“…Moreover, there are polynomial-time sequential algorithms to obtain optimal (that is, using a minimum number of colours) colourings [2,3,5,10,12]. NC is the class of problems solvable in polylogarithmic (i.e.0(logkn), for some k) parallel time with a polynomial number of processors.…”