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 2 1 1 4 1 2 2 1 3 2 3 2 2 1 1 2 Chromatic Index Chromatic Number ′ ( G ) = 3 χ χ ( G ) = 2 Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 3 / 14
k -total labeling 1 A total labeling is an assignment of positive 4 numbers to both vertices and edges, where 2 1 no two adjacent vertices share the same 3 1 3 1 4 label, 5 3 2 no two incident edges share the same 2 2 2 label, 4 1 3 and no incident edge and vertex share 3 the same label. Total Chromatic Number ′′ ( G ) = 5 χ Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 4 / 14
k -graceful labeling 3 5 1 A graceful labeling is obtained by 1 4 1 properly label the set of vertices 2 2 4 6 2 label the edges by taking the absolute 3 5 3 difference of incident vertex labels 1 1 3 where the set of edges are also properly 1 2 labeled Graceful Chromatic Number χ g ( 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. ( χ td ( 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 3 2 4 4 3 1 1 3 3 1 1 4 3 1 4 2 4 3 1 2 3 P 5 3 2 1 5 1 4 3 C 10 χ td ( P n ) = 4 for n ≥ 4 � 4 if n ≡ 0 (mod 3 ) χ td ( C n ) = 5 otherwise 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 3 1 2 5 8 1 2 2 3 3 ( 2 , 1 ) -double ( 1 , 3 , 1 ) -sequence ( 8 , 5 , 2 ) -sequence 2 Determine the smallest k such that G will be labeled this way. ( χ td ( G ) ) Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 8 / 14
Stars and Wheels 2 5 3 9 4 12 13 7 1 6 10 9 11 8 W 13 1 2 3 4 5 6 7 K 1 , 7 8 n = 4 � m + 1 , m is even 7 n = 5 χ td ( K 1 , m ) = χ td ( W n ) = m + 2 , m is odd n + 1 n is even and n ≥ 6 n is odd and n ≥ 7 n Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 9 / 14
Caterpillars G ∆ + 1 ≤ χ td ( 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 2 3 2 3 4 3 4 5 2 3 1 9 10 1 9 10 4 5 6 6 7 6 7 4 6 5 Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 13 / 14
Lobsters and Maximal Rooted Trees T 4 , 2 For a maximal rooted tree with height 2, � 3 ∆ + 3 � χ td ( T ∆ , 2 ) = . 2 H For any maximal rooted tree with height h , where h ≥ 2, ∆ + 1 ≤ χ td ( H ) ≤ ∆ 1 + ∆ 2 + 1 � 3 ∆ + 3 � ≤ χ td ( T ∆ , h ) ≤ 2 ∆ + 1 2 . Yufei Zhang (Saint Mary’s College) Puzzle on Graphs: Total Difference Labelings of Graphs February 2, 2020 14 / 14
Recommend
More recommend
