Let k ≥ 2 and let G be a graph of chromatic number k such that χ(G − {v}) < k for every v ∈ V (G).
a) if k = 2, 3, describe the graph G.
b) Prove that δ(G) ≥ k − 1.
c) Show that G is a block
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Let k ≥ 2 and let G be a graph of chromatic number k such that χ(G − {v}) < k for every v ∈ V (G). a) if k = 2, 3, de...
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1} such that c(u)≠c(v) for every edge (u, v) ∈ E. In other words, the numbers 0.1,... k-1 represent the k colors, and adjacent vertices different colors. must havec. Let d be the maximum degree of any vertex in a graph G. Prove that we can color G with d +1 colors.
Most Edges. Prove that if a graph with n vertices has chromatic number n, then the graph has n(n-1) edges. Divide. Let V = {1, 2, ..., 10} and E = {(x, y) : x, y € V, x + y, , and a divides y}. Draw the directed graph with vertices V and directed edges E.
show all the work
a9) What is meant by coloring the vertices of a graph? Define the chromatic number of a graph. a) What is the famous 4- color theorem? b) Translate the following map into a graph G and find χ (G). You may draw G embedded in the map or alongside the map. No need to consider the outside region. 了 jo
a9) What is meant by coloring the vertices of a graph? Define the chromatic number of...
Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let Δ(G) be the maximum degree of any vertex in G, δ(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that δ(G)n2≤m≤Δ(G)n2
Let G be a non-Hamiltonian, connected graph. For every pair of nonadjacent vertices u and v, 8(u) +8()2 k, for some k> O. Show that G contains a path of length k.
Let G be a non-Hamiltonian, connected graph. For every pair of nonadjacent vertices u and v, 8(u) +8()2 k, for some k> O. Show that G contains a path of length k.
(a) Find the chromatic polynomial PG(k) of the following graph G. Give 6. your answer in factorized form. (b) Write down x(G), the chromatic number of G (c) Without expanding Pc(k), find the coefficient of k28.
(a) Find the chromatic polynomial PG(k) of the following graph G. Give 6. your answer in factorized form. (b) Write down x(G), the chromatic number of G (c) Without expanding Pc(k), find the coefficient of k28.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
(5) Let G be a graph without loops. Let n be the number of vertices and let xG(z) be its chromatic polynomial. Recall from HW5 that 2(G) is the number of cycle-free orientations of G. Show that
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...