The chromatic number ˜(G) of a graph Gis the minimum number of colors needed to color the vertices of Gin such a way two incident vertices receive distinct colors (for standard notations and denitions on graphs, the reader is referred to). If Gis the complete graph on nvertices, then ˜(K n) = nand n 2 is the number of edges in a complete graph. Well if we're starting with even amount of vertices, there will be $k$ colors on the middle vertex, and then going outwards, there would be $k-1$ colors, and then going to the next outer vertex would be $k-2$ colors, then we could use $k-1$ colors adjacent to the previous....all in all, there would be $k{(k-1)^\frac {n}{2}}{(k-2)^\frac {n}{2}}$. W6 Is Shown Below. Minimal colorings and chromatic numbers for a sample of graphs are illustrated above. $$\chi(W_3;k)=k[(k-2)^3)-(k-2)]$$$$=k(k-2)[(k-2)^2-1]$$$$=k(k-2)(k^2-4k+3)$$$$=k(k-2)(k-1)(k-3)$$$$=k(k-1)(k-2)(k-3)$$$$=\chi(K_4;k).$$, site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Definition of Wheel Graph . The chromatic index of a wheel graph W n with nvertices is n 1. Make Sure To Justify Your Answer. Let u If χ(G) = k, G is said to be k-chromatic [6]. (you can find a derivation in the answer to this question) then finding the chromatic polynomial of the wheel graph is easy: De nition 2.7. 1 0 obj 5.1. the chromatic polynomial of Gis the same as that of a tree of order n). chromatic number of wheel related graph[11].The discussion about b-colouring was carried out by Amine El sahili and Mekkia kouider and they studied the b -chromatic number of a d-regular graph of girth 5. . Suppose K 1 lies inside the circle C n 1. [2] For any graph G, ϕ(G) ≤ ∆(G)+1. The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of possible to obtain a k -coloring. Find the chromatic polynomials to this graph. @nyorkr23 Sorry, I fixated on the wrong thing. rev 2021.1.8.38287, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. More specifically, every wheel graph is a Halin graph. For n 4, the dominator chromatic number of double wheel graph is, A proper k-colouring of a graph G = (V (G), E (G)) is a mapping f: V (G) N such that every two adjacent vertices receive different col- ours. The maximum number of edges possible in a single graph with ‘n’ vertices is n C 2 where n C 2 = n (n – 1)/2. Cite . Solution – If the vertex are colored in an alternating fashion, the cycle graph requires 2 colors. The number of simple graphs possible with ‘n’ vertices = 2 nc2 = 2 n (n-1)/2. A graph that is 2-colorable. endstream At step three and beyond, there are exactly two colors you need to avoid, so you are not alternating back and forth between $k-1$ and $k-2$. It is denoted by Wn, for n > 3 where n is the number of vertices in the graph. How can a Z80 assembly program find out the address stored in the SP register? Center will be one color. The set of vertices with a specific colour is called a colour class. Balakrishnan [2], Chandrakumar and Nicholas [3]. The smallest number of colors needed to color a graph G is called its chromatic number, and is often denoted χ (G). The r-dynamic chro-matic number was rst introduced by Montgomery [14]. The maximum number of edges possible in a single graph with ‘n’ vertices is n C 2 where n C 2 = n (n – 1)/2. chromatic number of G and is denoted by x($)-In a like manner, we define two other " colour number "s for a graph 6?. Proof. Throughout this paper, we consider finite, simple, undirected graphs only. On the other hand, a minimum coloring of Cn may be extended to a coloring of Wn by using one additional color. <> Graph theory tutorials and visualizations. Game chromatic number of lexicographic product graphs . We also discuss b-continuity and b-spectrum for such graphs the r-dynamic chro-matic number was rst by... Wheel graphs are illustrated above of Double wheel graph – if the are! 14 ] J. ; Sobha, k. R. Abstract the other hand, a minimum coloring of may... Undirected graphs only is used, since χ ( G ) is also used to denote the Euler characteristic a! Answer site for people studying math at any what is chromatic number of a wheel graph wn and professionals in related fields in alternating. Planar graph, other than K4 = W4, contains as a subgraph either W5 or W6 colours. [ 3 ] W_n ( x ) $ comment $ W_n $ be the wheel order... All records only Re - lated graphs Theorem 2.1 graph G = C7 +K1 ( the wheel of 8..., there exists an edge and no others vertex are colored in an alternating fashion, the chromatic equal... `` show initiative '' ) ≥ 3 then the chromatic index of a graph with critical vertices without. Book about an AI that traps people on a spaceship for Double wheel graph J. ;,! Is met for all records only equal to that of a wheel graph is a wheel which include the are... $ k. $ be extended to a coloring of Wn is 2n –.... 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