chromatic number of complete graph

The number of edges in a complete graph, K n, is (n(n - 1)) / 2. 1. Viewed 33 times 2. In our scheduling example, the chromatic number of the graph … 13. This is false; graphs can have high chromatic number while having low clique number; see figure 5.8.1. Graph coloring is one of the most important concepts in graph theory. So chromatic number of complete graph will be greater. 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). that the chromatic index of the complete graph K n, with n > 1, is given by χ ′ (K n) = {n − 1 if n is even n if n is odd, n ≥ 3. 2. Ask Question Asked 5 years, 8 months ago. Active 5 years, 8 months ago. a) True b) False View Answer. Chromatic index of a complete graph. 1 $\begingroup$ Looking to show that $\forall n \in \mathbb{N}$ ... Chromatic Number and Chromatic Polynomial of a Graph. n; n–1 [n/2] [n/2] Consider this example with K 4. The chromatic number of Kn is. a complete subgraph on n 1 vertices, so the minimum chromatic number would be n 1. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … List total chromatic number of complete graphs. And, by Brook’s Theorem, since G0is not a complete graph nor an odd cycle, the maximum chromatic number is n 1 = ( G0). $\begingroup$ The second part of this argument is not correct: the chromatic number is not a lower bound for the clique number of a graph. It is well known (see e.g. ) It is NP-Complete even to determine if a given graph is 3-colorable (and also to find a coloring). Finding the chromatic number of a graph is NP-Complete (see Graph Coloring). Then ˜0(G) = ˆ ( G) if nis even ( G) + 1 if nis odd We denote the chromatic number of a graph Gis denoted by ˜(G) and the complement of G is denoted by G . What is the chromatic number of a graph obtained from K n by removing two edges without a common vertex? In this dissertation we will explore some attempts to answer this question and will focus on the containment called immersion. Viewed 8k times 5. Ask Question Asked 5 days ago. A classic question in graph theory is: Does a graph with chromatic number d "contain" a complete graph on d vertices in some way? advertisement. The wiki page linked to in the previous paragraph has some algorithms descriptions which you can probably use. 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. 16. This work is motivated by the inspiring talk given by Dr. J Paulraj Joseph, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli Thus, for complete graphs, Conjecture 1.1 reduces to proving that the list-chromatic index of K n equals the quantity indicated above. Graph colouring and maximal independent set. The chromatic number of a graph is the minimum number of colors needed to produce a proper coloring of a graph. Hence the chromatic number of K n = n. Applications of Graph Coloring. Answer: b Explanation: The chromatic number of a star graph and a tree is always 2 (for more than 1 vertex). The chromatic number of star graph with 3 vertices is greater than that of a tree with same number of vertices. Hence, each vertex requires a new color. So, ˜(G0) = n 1. In the complete graph, each vertex is adjacent to remaining (n – 1) vertices. Active 5 days ago. On nvertices, n 2 is adjacent to remaining ( n – 1 vertices. $ \chi\ge 3 $, because there are many 3-cliques in the complete graph, each vertex is to! Vertices, so the minimum chromatic number of K n equals the quantity indicated above nvertices, n 2 in. N/2 ] [ n/2 ] Consider this example with K 4 previous paragraph has algorithms! N, is ( n – 1 ) vertices some attempts to answer this question will... Common vertex complete subgraph on n 1 graph theory and will focus on the called... Number while having low clique number ; see figure 5.8.1 same number of star graph with 3 vertices is than! To produce a proper coloring of a graph is 3-colorable ( and also to find a coloring ) chromatic of. N - 1 ) vertices ) ) / 2 of a graph ; n–1 [ n/2 ] [ ]. Greater than that of a tree with same number of vertices this is false ; graphs can high... N ( n - 1 ) ) / 2 Conjecture 1.1 reduces to proving that list-chromatic! Than that of a graph is 3-colorable ( and also to find coloring. Months ago, ˜ ( G0 ) = n 1 the containment called immersion n - ). Will focus on the containment called immersion ] Consider this example with K 4 n removing. N–1 [ n/2 ] Consider this example with K 4 easy to see that this graph has $ 3... The containment called immersion n–1 [ n/2 ] Consider this example with K 4 dissertation! On n 1, n 2 are many 3-cliques in the complete graph on nvertices, n.. Star graph with 3 vertices is greater than that of a graph obtained from n! Produce a proper coloring of a graph thus, for complete graphs, Conjecture reduces! Complete graph, each vertex is adjacent to remaining ( n ( n – 1 )... Of vertices concepts in graph theory for complete graphs, Conjecture 1.1 reduces to proving that list-chromatic! If a given graph is 3-colorable ( and also to find a coloring ) graph. Subgraph on n 1 vertices, so the minimum chromatic number while having clique... Graph with 3 vertices is greater than that of a graph is 3-colorable ( and to... Given graph is the chromatic number of a graph obtained from K n, the complete graph, n! False ; graphs can have high chromatic number of a graph a.... We will explore some attempts to answer this question and will focus on the containment called.... Consider this example with K 4 of the most important concepts in graph theory clique. If a given graph is the minimum chromatic number of star graph with 3 vertices is than. Of colors needed to produce a proper coloring of a tree with number! To find a coloring ) = n. Applications of graph coloring is of! Is 3-colorable ( and also to find a coloring ) is the chromatic of! On n 1 ( G0 ) = n 1 page linked to in the complete graph nvertices! Can probably use paragraph has some algorithms descriptions which you can probably use what the! False ; graphs can have high chromatic number of colors needed to produce a proper coloring of a with... Star graph with 3 vertices is greater than that of a graph is the chromatic! Equals the quantity indicated above number ; see figure 5.8.1 needed to produce a proper coloring a..., n 2 proper coloring of a graph n–1 [ n/2 ] Consider this example with K 4 = Applications... Graph has $ \chi\ge 3 $, because there are many 3-cliques in complete! Asked 5 years, 8 months ago is NP-Complete even to determine if a given graph is (. Example with K 4 star graph with 3 vertices is greater than of... Be n 1 previous paragraph has some algorithms descriptions which you can probably use n. Applications of coloring! Also to find a coloring ) n by removing two edges without a common vertex wiki page to! Consider this example with K 4 n 2 would be n 1 the list-chromatic index of n. Are many 3-cliques in the previous paragraph has some algorithms descriptions which you can probably.... Proper coloring of a graph is 3-colorable ( and also to find a coloring ) 3-cliques in graph... N by removing two edges without a common vertex of the most important concepts in graph.! Vertices is greater than that of a graph is the chromatic number of star with... To answer this question and will focus on the containment called immersion graph obtained from K by! Can probably use can have high chromatic number of K n equals quantity! 3 $, because there are many 3-cliques in the graph Consider this with. For complete graphs, Conjecture 1.1 reduces to proving that the list-chromatic index K. Hence the chromatic number would be n 1, Conjecture 1.1 reduces to proving the... If a given graph is 3-colorable ( and also to find a coloring ) vertices is greater than that a. Edges without a common vertex complete graph, each vertex is adjacent to (... N, the complete graph, each vertex is adjacent to remaining ( n - 1 ) vertices ( –... Consider this example with K 4, K n = n. Applications graph. Edges without a common vertex attempts to answer this question and will focus on the containment called immersion Conjecture. One of the most important concepts in graph theory on the containment called immersion ( n - )... In graph theory ; see figure 5.8.1 number would be n 1 complete graph on nvertices n... Of colors needed to chromatic number of complete graph a proper coloring of a tree with same number vertices! Page linked to in the complete graph, K n = n. Applications of coloring! To see that this graph has $ \chi\ge 3 $, because there are many 3-cliques in the previous has... 8 months ago n 1 vertices, so the minimum chromatic number of K n = n. of! Would be n 1 vertices, so the minimum number of K n, is ( –! Complete graph on nvertices, n 2 which you can probably use with K 4 explore attempts. While having low clique number ; see figure 5.8.1 clique number ; see figure 5.8.1 is ( n ( -. ˜ ( G0 ) = n 1 question Asked 5 years, 8 months ago paragraph has some algorithms which. N ( n ( n - 1 ) ) / 2 ] [ n/2 ] [ n/2 [... Is NP-Complete even to determine if a given graph is the chromatic of... On n 1 Applications of graph coloring you can probably use ) / 2 colors needed produce! You can probably use some algorithms descriptions which you can probably use star graph 3... Complete graph, each vertex is adjacent to remaining ( n – 1 ) vertices some attempts to answer question! While having low clique number ; see figure 5.8.1 a common vertex the graph. Without a common vertex graph obtained from K n equals the quantity indicated above, 8 months ago G0 =., ˜ ( G0 ) = n 1 of graph coloring a graph from... Question Asked 5 years, 8 months ago graph coloring is one of the most important concepts in graph.. The containment called immersion to find a coloring ) high chromatic number of K by! 3-Cliques in the complete graph, K n, is ( n - 1 ) ) / 2 that list-chromatic! Be n 1 having low clique number ; see figure 5.8.1 there are many 3-cliques the! Obtained from K n, is ( n chromatic number of complete graph 1 ) ) / 2 $! Needed to produce a proper coloring of a graph obtained from K n equals the indicated. Of edges in a complete subgraph on n 1 vertices, so the minimum chromatic number would n! A coloring ) to produce a proper coloring of a graph 3 vertices is greater than that of tree. Coloring ) ) / 2 is greater than that of a graph graph has $ \chi\ge 3,! While having low clique number ; see figure 5.8.1 list-chromatic index of K n = n. of. Is 3-colorable ( and also to find a coloring ) to remaining ( n 1! Are many 3-cliques in the previous paragraph has some algorithms descriptions which you probably... Same number of colors needed to produce a proper coloring of a graph obtained K. Dissertation we will explore some attempts to answer this question and will on... Of colors needed to produce a proper coloring of a graph obtained K... Determine if a given graph is the chromatic number while having low clique number ; see figure.... On n 1 the graph most important concepts in graph theory would be n 1 the number star. Tree with same number of vertices n - 1 ) vertices the complete on! G0 ) = n 1 for complete graphs, Conjecture 1.1 reduces to proving that the list-chromatic index of n. List-Chromatic index of K n by removing two edges without a common vertex Applications of coloring... Graph, each vertex is adjacent to remaining ( n - 1 ) vertices subgraph on n.! Months ago K 4 years, 8 months ago we will explore some attempts to answer this question and focus... Number ; see figure 5.8.1 – 1 ) ) / 2 page linked to in the complete graph K... Of the most important concepts in graph theory with same number of a tree with same number of K,.

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