On 3-colourable K 4. -free graphs. M. A. Shalu Department of Mathematics Indian Institute of Technology, Madras

Transcription

1 On 3-colourable K 4 -free graphs M. A. Shalu Department of Mathematics Indian Institute of Technology, Madras

2 1 ORGANISATION PRELIMINARIES LITERATURE SURVEY WORK DONE WORK IN PROGRESS REFERENCES VISIBLE RESEARCH OUTPUT

3 2 PRELIMINARIES Clique : A set of vertices which are pairwise adjacent.

4 2 PRELIMINARIES Clique : A set of vertices which are pairwise adjacent.

5 2 PRELIMINARIES Clique : A set of vertices which are pairwise adjacent. ω(g) =: Size of the largest clique in G

6 2 PRELIMINARIES Clique : A set of vertices which are pairwise adjacent. ω(g) =: Vertex colouring : Size of the largest clique in G No two adjacent vertices have same colour

7 3 LITERATURE SURVEY C. E. Shannon [1956] : Characterize the class G with χ(g) = ω(g) for G G

8 3 LITERATURE SURVEY C. E. Shannon [1956] : Characterize the class G with χ(g) = ω(g) for G G P. Erdos [1959] : For any two integers g 3 and k 3

9 3 LITERATURE SURVEY C. E. Shannon [1956] : Characterize the class G with χ(g) = ω(g) for G G P. Erdos [1959] : For any two integers g 3 and k 3 there exists a graph G with χ(g) = k and girth g

10 3 LITERATURE SURVEY C. E. Shannon [1956] : Characterize the class G with χ(g) = ω(g) for G G P. Erdos [1959] : For any two integers g 3 and k 3 there exists a graph G with χ(g) = k and girth g = There is no upper bound for the chromatic number as a function (known as χ -binding function) of its clique number of an arbitrary graph

11 4 χ(g) ω(g) + 1

12 4 χ(g) ω(g) + 1 Line graphs

13 4 χ(g) ω(g) + 1 Line graphs {K 1,3, K 5 e} -free graphs (Kierstead 1984)

14 4 χ(g) ω(g) + 1 Line graphs {K 1,3, K 5 e} -free graphs (Kierstead 1984) {K 1,3, (K 2 K 1 ) + K 2 } -free graphs (Medha 1989)

15 5 I. Holyer [1981] Determine the chromatic number of {K 1,3, K 4 }-free 4-regular graphs is an N P -complete problem.

16 5 I. Holyer [1981] Determine the chromatic number of {K 1,3, K 4 }-free 4-regular graphs is an N P -complete problem. Problem : Identify subclasses of {K 1,3, K 4 }-free graphs with χ(g) = ω(g).

17 6 WORK DONE 4-critical {K 1,3, K 4 }-free graph :

18 6 WORK DONE 4-critical {K 1,3, K 4 }-free graph : χ(g) = 4.

19 6 WORK DONE 4-critical {K 1,3, K 4 }-free graph : χ(g) = 4. χ(g S) < χ(g) for every non-empty set S V (G).

20 6 WORK DONE 4-critical {K 1,3, K 4 }-free graph : χ(g) = 4. χ(g S) < χ(g) for every non-empty set S V (G). G is {K 1,3, K 4 }-free.

21 6 WORK DONE 4-critical {K 1,3, K 4 }-free graph : χ(g) = 4. χ(g S) < χ(g) for every non-empty set S V (G). G is {K 1,3, K 4 }-free.

22 7

23 7

24 7

25 8 RESULTS Theorem A : If G ( C 2n+1,n 2) is a {K 1,3, K 1 + C 5, L, M}-free connected graph, then χ(g) = ω(g).

26 8 RESULTS Theorem A : If G ( C 2n+1,n 2) is a {K 1,3, K 1 + C 5, L, M}-free connected graph, then χ(g) = ω(g).

27 8 RESULTS Theorem A : If G ( C 2n+1,n 2) is a {K 1,3, K 1 + C 5, L, M}-free connected graph, then χ(g) = ω(g). Theorem B: If G ( C 2n+1, n 2) is a {K 1,3, K 1 + C 5, M, N, P }-free connected graph, then χ(g) = ω(g).

28 8 RESULTS Theorem A : If G ( C 2n+1,n 2) is a {K 1,3, K 1 + C 5, L, M}-free connected graph, then χ(g) = ω(g). Theorem B: If G ( C 2n+1, n 2) is a {K 1,3, K 1 + C 5, M, N, P }-free connected graph, then χ(g) = ω(g).

29 9 Contd... Theorem C: If G ( C 2n+1, n 2) is a {K 1,3, K 1 + C 5, K 4, P, P, R 1, R 2, R 3, S}-free connected graph, then χ(g) = ω(g).

30 10 WORK IN PROGRESS To study the chromatic number of {K 1,3, K 1 + C 4 }-free graphs.

31 10 WORK IN PROGRESS To study the chromatic number of {K 1,3, K 1 + C 4 }-free graphs. To find 4-critical {K 1,3, K 4 }-free graphs with minimum number of vertices.

32 11 References [1] Berge,C. Chvatal, V.: In: Topics on Perfect Graphs. Annals of Discrete Mathematics, Vol 21 Amsterdam: North -Holland (1984) [2] Beineke, L. W.: Derived graphs and digraphs.in: H. Sachs et al. Beitrage zur Graphentheoria, Teubner Leipzig 17-33(1968) [3] Brandt, S.: Triangle-free graphs and forbidden subgraphs. Discrete App. Math. 120, 25-33(2002) [4] Erdös, P.: Graph theory and probability II. Canad. J. Math. 13, (1961) [5] Fiorini, S.: On the chormatic index of outerplanar graphs. J. Combin. Theory Ser. B 18, 35-38(1975).

33 12 [6] Gyárfás, A.: On Ramsey covering numbers. In: A. Hajnal et al.: Infinite and Finite Sets, Colloq. Math. Soc. Jànos Bolyai 10, (1975). [7] Holyer, I.: The N P -completeness of edge-colouring. SIAM. J. Comput. 10(4) (1981). [8] Kierstead, H.: On the chromatic index of multigraphs without large triangles. J. Combin. Theory Ser.B 36, (1984) [9] Medha, D.: On the chromatic number of a graph with two forbidden subgraphs. J. Combin. Theory Ser. B 46, 1-6 (1989). [10] Mycielski, J.: Sur le coloriage des graphes. Colloq.Math. 3, (1955). [11] Shannon, C. E.: The zero error capacity of a noisy channel, IRE Trans. Inform. Theory IT (1956)

34 13 [12] Randerath, B.: The Vizing bound for the chromatic number based on forbidden pairs. Ph.D. Thesis, RWTH Aachen: Shaker Verlag 1998 [13] Vizing, V. G.: On an estimate of the chromatic number of a p-graph. (Russian) Diskret. Analiz. 3, 25-30(1964) [14] Vizing, V.G.: The chromatic class of a multigraph, Kibernetika (Kiev) 1, 29-39(1965); [Russian] English translation in Cybernetics 1, 32-41(1965) [15] Yap, H. P.: Some Topics in Graph Theroy. London Mathematical Society (Lect. Note Series. 108) 1986

35 14 VISIBLE RESEARCH OUTPUT 1. S A Choudum, Shalu M A, Dissolved Graphs and Strong Perfect Graph Conjecture, Ars Combinatoria (Accepted for publication). 2. S A Choudum, Shalu M A, A class of graphs with clique number and chromatic number three, Graphs and Combinatorics (Communicated). 3. S A Choudum, Shalu M A, A class of 3-colourable K 4 -free graphs, Discrete Mathematics (Communicated).