It is easy to see that this graph has $\chi\ge 3$, because there are many 3-cliques in the graph. n, the complete graph on nvertices, n 2. Finding the chromatic number of a graph is NP-Complete (see Graph Coloring). Graph coloring is one of the most important concepts in graph theory. The number of edges in a complete graph, K n, is (n(n - 1)) / 2. This work is motivated by the inspiring talk given by Dr. J Paulraj Joseph, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli Viewed 8k times 5. A classic question in graph theory is: Does a graph with chromatic number d "contain" a complete graph on d vertices in some way? So, Ë(G0) = n 1. $\begingroup$ The second part of this argument is not correct: the chromatic number is not a lower bound for the clique number of a graph. Then Ë0(G) = Ë ( G) if nis even ( G) + 1 if nis odd We denote the chromatic number of a graph Gis denoted by Ë(G) and the complement of G is denoted by G . Hence the chromatic number of K n = n. Applications of Graph Coloring. that the chromatic index of the complete graph K n, with n > 1, is given by Ï â² (K n) = {n â 1 if n is even n if n is odd, n â¥ 3. a) True b) False View Answer. 13. In this dissertation we will explore some attempts to answer this question and will focus on the containment called immersion. The chromatic number of Kn is. n; nâ1 [n/2] [n/2] Consider this example with K 4. It is NP-Complete even to determine if a given graph is 3-colorable (and also to find a coloring). 1. List total chromatic number of complete graphs. 2. An example that demonstrates this is any odd cycle of size at least 5: They have chromatic number 3 but no cliques of size 3 (or larger). So chromatic number of complete graph will be greater. The chromatic number of a graph is the minimum number of colors needed to produce a proper coloring of a graph. 1 $\begingroup$ Looking to show that $\forall n \in \mathbb{N}$ ... Chromatic Number and Chromatic Polynomial of a Graph. Active 5 days ago. It is well known (see e.g. ) The wiki page linked to in the previous paragraph has some algorithms descriptions which you can probably use. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share â¦ And, by Brookâs Theorem, since G0is not a complete graph nor an odd cycle, the maximum chromatic number is n 1 = ( G0). Answer: b Explanation: The chromatic number of a star graph and a tree is always 2 (for more than 1 vertex). Ask Question Asked 5 days ago. What is the chromatic number of a graph obtained from K n by removing two edges without a common vertex? This is false; graphs can have high chromatic number while having low clique number; see figure 5.8.1. advertisement. Hence, each vertex requires a new color. 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