Puzzle on Graphs: Total Difference Labelings of Graphs Yufei Zhang - - PowerPoint PPT Presentation

Puzzle on Graphs: Total Difference Labelings of Graphs

Yufei Zhang

Saint Mary’s College Department of Math and Computer Science

February 2, 2020

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 1 / 14

Graph

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 2 / 14

Proper Vertex Labeling and Proper Edge Labeling

1 2 1 2 1 2 2 Chromatic Number χ(G) = 2 1 1 1 2 2 2 3 3 4 Chromatic Index χ

′(G) = 3 Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 3 / 14

k-total labeling

A total labeling is an assignment of positive numbers to both vertices and edges, where

1 no two adjacent vertices share the same

label,

2 no two incident edges share the same

label,

3 and no incident edge and vertex share

the same label. 4 1 4 3 4 2 5 2 1 1 1 3 2 3 2 3 Total Chromatic Number χ

′′(G) = 5 Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 4 / 14

k-graceful labeling

A graceful labeling is obtained by

1 properly label the set of vertices 2 label the edges by taking the absolute

difference of incident vertex labels

3 where the set of edges are also properly

labeled 4 3 1 2 5 1 6 1 4 1 2 2 3 3 5 1 Graceful Chromatic Number χg(G) = 6

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 5 / 14

Total Difference Labeling

Let G be a graph.

Steps:

1 Label the vertices with any number in the set of {1, 2, . . . , k}.

1 4 5 1 3

2 Label the edges with the absolute difference of end vertex labels.

1 4 5 1 3 3 1 4 2

3 Make sure the labeling of G forms a total labeling.

1 2 3 1 4 1 1 2 3

4 Determine the smallest k such that G will be labeled this way. (χtd(G))

1 4 3 1 4 3 1 2 3

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 6 / 14

Paths and Cycles

1 4 3 1 4 3 1 2 3 P5 χtd(Pn) = 4 for n ≥ 4 1 3 1 3 1 3 4 1 4 1 4 1 3 2 3 2 3 2 5 4 C10 χtd(Cn) = 4 if n ≡ 0 (mod 3) 5

• therwise

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 7 / 14

Total Difference Labeling

Let G be a graph.

Steps:

1 Properly label the vertices with any number in the set of {1, 2, . . . , k}, where the vertex

labels do not contain doubles or 3-sequences. 1 2 1 (2, 1)-double 1 3 1 2 2 (1, 3, 1)-sequence 2 5 8 3 3 (8, 5, 2)-sequence

2 Determine the smallest k such that G will be labeled this way. (χtd(G)) Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 8 / 14

Stars and Wheels

9 1 2 3 4 5 6 7 K1,7 χtd(K1,m) = m + 1, m is even m + 2, m is odd 13 1 3 5 7 9 11 2 4 6 8 10 12 W13 χtd(Wn) =        8 n = 4 7 n = 5 n + 1 n is even and n ≥ 6 n n is odd and n ≥ 7

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 9 / 14

Caterpillars

G ∆ + 1 ≤ χtd(G) ≤ ∆ + 3

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 10 / 14

∆ + 3

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 11 / 14

∆ + 3

1 9 10 1 9 10

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 12 / 14

∆ + 3

1 9 10 1 9 10 2 3 2 3 4 3 4 5 2 3 4 5 6 6 7 6 7 4 5 6

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 13 / 14

Lobsters and Maximal Rooted Trees

H ∆ + 1 ≤ χtd(H) ≤ ∆1 + ∆2 + 1 T4,2 For a maximal rooted tree with height 2, χtd(T∆,2) = 3∆ + 3 2

• .

For any maximal rooted tree with height h, where h ≥ 2, 3∆ + 3 2

• ≤ χtd(T∆,h) ≤ 2∆ + 1

.

Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 14 / 14