Validity of Borodin & Kostochka Conjecture for a Class of Graphs

Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of {\omega} and maximum degree minus 1. Here we prove that this Conjecture is true for {P3 UNION K1}-free graphs and {K2 UNION complement of K2}-free graphs.Comment: 3 page

On the Tight Chromatic Bounds for a Class of Graphs without Three Induced Subgraphs

Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden subgraphs is necessary in the hypothesis.Comment: 4 pages, 2 figure

Forbidden Subgraph Characterization of Quasi-line Graphs

Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques.Comment:

Validity of Borodin and Kostochka Conjecture for {4 Times K1}-free Graphs

Problem of finding an optimal upper bound for the chromatic no. of even 3K1-free graphs is still open and pretty hard. Here we prove Borodin & Kostochka Conjecture for 4K1-free graphs G i.e. If maximum degree of a {4 Times K1}-free graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of {\omega} and {\delta-1}.Comment: 5 page

Tight Chromatic Upper Bound for {3K1, K1+C4}-free Graphs

Problem of finding an optimal upper bound for the chromatic no. of 3K1-free graphs is still open and pretty hard. It was proved by Choudum et al that an upper bound on the chromatic no. of {3K1, K1+C4}-free graphs, is 2{\omega}. We improve this by proving that if G is {3K1, K1+C4}-free, then its chromatic no. is less than or equal to 3{\omega} divided by 2, where {\omega} is the size of a maximum clique in G. Also we give examples to show that this bound is tight.Comment: 2 page

On Chromatic no. of 3K1-free graphs and R(3, k)

Here we prove that if G has independence no. 2 and clique size omega with omega less than or equal to 11, then (1) chromatic no. is less than or equal to (omega2+12omega-13)/8, if omega is odd, and (2) chromatic no. is less than or equal to (omega2+10omega)/8, if omega is even. We further conjecture that the results are true in general for all omega. We also conjecture that (A) if omega is odd and R(3, omega) is even, then R(3, omega) = (omega2+8omega-9)/4, (B) if omega and R(3, omega) are both odd, then (omega2+8omega-13)/4, (C) if omega and R(3, omega) are both even, then R(3, omega) = (omega2+6omega)/4 and (D) if omega is even and R(3, omega) is odd, then (omega2+6omega-4)/4. Again we verify the results for omega less than or equal to 9.Comment: 5 Page

Tight Chromatic Upper Bound for {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free Graphs

Problem of finding an optimal upper bound for {\chi} of (3 Times K1)-free graphs is still open and pretty hard. It was proved by Choudum et al that upper bound on the {\chi} of {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free graphs is 2{\omega}. We improve this by proving that if G is {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free, then {\chi} less than or equal to 3{\omega} divided by 2 for {\omega} not equal to 5, and {\chi} less than or equal to 8 for {\omega} = 5 where {\omega} is the size of a maximum clique in G. We also give examples of extremal graphs.Comment: 7 page

Tight Chromatic Number for a class of Graphs with two forbidden subgraphs

Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} - 3, {\chi} equals its maximum clique size. We also give examples to show that the condition is necessary.Comment: 3 page

Improvement on Brook theorem for (3 Times K1)-free Graphs

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less than or equal to max (maximum degree-1, {\omega}). We also prove that if G is (3 Times K1)-free, {\omega} is equal to 4 and maximum degree is greater than or equal to 7, then {\chi} is less than or equal to maximum degree-1. This implies that Borodin & Kostochka Conjecture is true for (3 Times K1)-free graphs as a corollary.Comment: 4 page

Validity of Borodin and Kostochka Conjecture for classes of graphs without a single, forbidden subgraph on 5 vertices

Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin & Kostochka Conjecture for (1) (P4 Union K1)-free (2) P5-free (3) Chair-free graphs and 4) graphs with dense neighbourhoods. Certain known results follow as Corollaries.Comment: 6 pages. arXiv admin note: text overlap with arXiv:1801.0131