Planar Graph Chromatic Number- Chromatic Number of any planar graph is always less than or equal to 4. gave a formula for the minimum size of a matching among k-regular (k − 2)-edge-connected graphs with a fixed number of vertices (see also ). EXERCISE: Draw two 3-regular graphs with six vertices. The graph Gis called k-regular for a natural number kif all vertices have regular degree k. Graphs that are 3-regular are also called cubic. Lemma 1 (Handshake Lemma, 1.2.1). In a complete graph, every pair of vertices is connected by an edge. Solution- Given-Number of edges = 24; Degree of each vertex = k . To see that this is tight, start with a graph on two vertices and one edge. If A⊆ E(G), then G[A] is the subgraph of Ginduced by A. In this paper, we present an algorithm to solve this problem for all k. bidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. The first interesting case is therefore 3-regular graphs, which are called cubic graphs (Harary 1994, pp. In a partial k-colouring of G, each edge of Gis Handshaking Theorem: We can say a simple graph to be regular if every vertex has the same degree. The problem of finding k-edge-connected components is a fundamental problem in computer science. Answer: b What is the possible biggest and the smallest number of edges in a graph with N vertices and K components? 20; 15; 10; 8 . 7. Introduction The concept of k-ordered graphs was introduced in 1997 by Ng and Schultz . 1.10 Give the set of edges and a drawing of the graphs K 3 [P 3 and K 3 P 3, assuming that the sets of vertices of K 3 and P 3 are disjoint. If G =((A,B),E) is a k-regular bipartite graph (k ≥ 1), then G has a perfect matching. Let’s start with a simple definition. 05, Apr 19. Property-02: Discrete Mathematics 48 (1984) 197-204 197 North-Holland REGULAR GRAPHS AND EDGE CHROMATIC NUMBER R.J. FAUDREE Memphis State University, Memphis, TN38152, USA J. SHEEHAN University of Aberdeen, The Edward Wright Building, Aberdeen, UK Received 23 September 1982 Revised 12 April 1983 For any simple graph G, Vizing's Theorem  implies that A (G)~)((G)<~ A(G)+ 1, where A … For example, if k is large enough, then we have C(G) < C(H) for any k-regular G and 1.001k-regular H on the same number of vertices. If n is the number of vertices of G, then the number of edges in a k-regular graph is nk/2. Is this correct? The degree d(v) of a vertex vis the number of edges that are incident to v. I think it also may depend on whether we have and even or an odd number of vertices? Likewise, a complete d-regular bipartite graph is denoted by Kd,d, having a number of vertices V(Kd,d) = 2d(i.e., dvertices of each of the two types), and a number of edges E(Kd,d) = d2. In a simple graph, the number of edges is equal to twice the sum of the degrees of the vertices. Prove that a k-regular graph of girth 4 has at least 2kvertices. A complete graph K n is a regular of degree n-1. An independent set of an undirected graph Gis a subset of its vertices such that none of the In graph theory, graphs can be categorized generally as a directed or an undirected graph.In this section, we’ll focus our discussion on a directed graph. Following are some regular graphs. So, to solve the problem, build A and construct A^k using matrix multiplication (the usual trick for doing exponentiation applies here). A trail is a walk with no repeating edges. A chordless path is a path without chords. On the other hand if no vivj, 2 6i