0
votes
1answer
34 views

Hanoi Algorithm With Different Nodes

http://en.wikipedia.org/wiki/Tower_of_Hanoi I need help developing a Hanoi algorithm which follows the same rules as the standard game, however the nodes that are transversed is different. In this ...
0
votes
2answers
29 views

Minimum cut in a graph does not change when the weight of all edges is increased by one

Suppose we have a Graph $G$ in which weight of all edges is $> 1$ (positive). If we increase weight of all edges by one, why does the minimum cut $(S, T)$ of $G$ into two graphs remain the same? ...
3
votes
0answers
34 views

A probability of a monochromatic cycle on a randomly colored lattice graph.

Let $G$ be an undirected $6 \times 6$ lattice graph. The $36$ vertices of $G$ are each randomly colored with one of $5$ colors with equal probability. Such a coloring is called "successful" if and ...
0
votes
2answers
30 views

Prove that $G$ is Hamiltonian.

Let $G=(V,E)$ be a connected graph which is not a tree. Prove that if for every cycle $C$ of the graph G and for any $v \in V(G)- V(C)$ there are more than $\frac{|C|}{2}$ edges from $v$ to $V(C)$ ...
-1
votes
1answer
47 views

The union of two connected graphs is connected [closed]

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...
0
votes
2answers
160 views

Help showing that every walk of length $k$ from $x$ to $y$ in a graph is a path.

If I were to suppose $x$ and $y$ are two vertices in the same connected component of a graph, and let $k$ be the distance between them, how would I prove that every walk of length $k$ from $x$ to $y$ ...
0
votes
1answer
37 views

Width and height of binary tree is $\theta(n)$?

we know this definition: Given a binary tree, Width of a tree is maximum of widths of all levels. Let us consider the below example tree. ...
0
votes
3answers
36 views

Proof verification: a connected graph always has a vertex that is not a cut vertex

Prove that every connected graph has vertices that even when you remove them, the graph stays connected. Let's assume that $\delta(G)>1$ becuase if it is equal to 1, the proof is trivial. I will ...
1
vote
2answers
42 views

Graph Degree and Some Condition

If $G$ be a Tree with degree $(5,r,s,1,1,1,1,1) $. (I wrote degree in non-increasing order). why all of this condition is True sometimes (I means on some condition)? I try to find an example that ...
0
votes
1answer
255 views

proof of a theorem in a paper

I was reading a paper named Decompositions of the Kronecker product of a cycle and a path into long cycles and long paths by P. K. Jha (Indian J. pure appl. Math. 23(8): 585-606, August 1992). In one ...
5
votes
3answers
364 views

Recurrence with varying coefficient

Problem 1 $$ {\rm f}\left(n\right) = \frac{1}{n}\, \left[{\rm f}\left(n - 1\right)k_{0} + {\rm f}\left(n-2\right)k_{1}\right]\tag{1} $$ ( This can also be written as ${\rm Q}\left(n\right) = ...
2
votes
1answer
18 views

Eccentricity of vertices in a graph when eccentricity of one vertex is given

I have a very basic doubt. If a vertex in any graph has the eccentricity two, then what can be concluded about eccentricities of other vertices in graph. Is the eccentricity of every vertex is less ...
0
votes
1answer
23 views

The “sides” of a k-regular bipartite graph are equal?

I was reviewing some lectures notes and noticed that in a proof of a theorem our lecturer stated that the "sides" of a k-regular bipartite graph are equal and that it is trivial to prove it. Anyway ...
0
votes
2answers
42 views

Graph Theory, with algorithms like kruskal and something more

The new government of the archipelago of Sealand has decided to join six islands by bridges to connect them directly. The cost of building a bridge depends on the distance between the islands. This ...
0
votes
1answer
24 views

Generators Trees in a Tree

My question is very short: How many spanning trees have a tree? Thanks in advance
1
vote
1answer
54 views

Stable Marriage - set of preferences such that every arrangement is stable?

This is a homework problem from the MIT OCW math for CS class, assignment 4, problem 5. Prove or disprove the following claim: for some n ≥ 3 (n boys and n girls, for a total of 2n people), there ...
6
votes
1answer
47 views

Does $K_{15,15}$ decompose into $K_{5,5}-C_{10}$ and $K_{5,5}-(C_6 \cup C_4)$ subgraphs?

Following on from this question: Q: Does $K_{15,15}$ decompose into $K_{5,5}-C_{10}$ and $K_{5,5}-(C_6 \cup C_4)$ subgraphs? or equivalently Q: Does there exist a $15 \times 15$ matrix ...
1
vote
1answer
52 views

Film Festival, with intersections graphs

I encourage you to read this problem. I have a doubt, have films 1 and 2 the same type? I read the problem and I think that films {1,3,5}, {2,4,6}, {3,4} and {5,6} are grouped, but not is the case ...
1
vote
1answer
26 views

Draw a graphic only passing one time

I would like to know when I can draw a graph, without lifting the pencil and passing once for each edge? What theory is behind that? Thanks for your time
0
votes
1answer
45 views

Making a Graph having edges

V is the set of those two-letter words built over {w, x, y, z} whose first letter is y or z. The graph G = (V, A) is defined so that two words of V determine an edge of A if they differ in exactly one ...
5
votes
2answers
92 views

Does $K_{12,12}$ decompose into $K_{4,4}-I$ subgraphs?

Q: Does the complete bipartite graph $K_{12,12}$ decompose into $K_{4,4}-I$ subgraphs, where $I$ is a $1$-factor (i.e., a perfect matching)? The obvious necessary conditions work: $K_{12,12}$ ...
4
votes
1answer
44 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
2
votes
1answer
45 views

Partition Graph Challenging Question

I want to find in which of the following Graph, the edges cannot partitioned to triangles? Km,n,r means 3-Partite Complete Graph with m, n, and r sections. a) K7 b) K12 c) K3,3,3 d) K5,5,5 i ...
0
votes
1answer
18 views

Planner Combination Problem on Graph

I ran into a Graph Problem. Suppose G is A Planner Graph with 100 Vertices such that if connect each two Non-adjacent vertices, the resulting graph would be non-planner. what is the number of edges ...
2
votes
1answer
45 views

Perfect Matching Combination Problem

We know: A perfect matching (a.k.a. 1-factor) is a matching which matches all vertices of the graph. if we remove edges of perfect matching of a 12-Complete Graph. how many triangle remain in this ...
1
vote
1answer
36 views

Convert adjacency matrix to graph

Is there any online service that can provide possible graphs (the simplest one) when I give a sequence of integers (node degrees) as input (or reject the input) -based on Erdős-Gallai formula? Thanks ...
0
votes
0answers
19 views

$k$ edge-disjoint $r$-arborescences in an acylic digraph

An $r$-arborescence of a digraph $D$ is a rooted spanning tree with root $r\in V(D)$ in which all the edges of $D$ are directed away from $r$. I would like to prove the following: I have thought ...
1
vote
2answers
52 views

Prove that a complement graph of a tree is either connected or it's a union of an isolated vertex and a full graph

I managed to prove the second part - that a tree that is one vertex with n-1 degree and all the rest are connected to it - the complement graph of such tree is an isolated vertex and the rest of the ...
1
vote
0answers
14 views

about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
14
votes
2answers
289 views

Does every “balloon” (dragon, tadpole, canoe paddle) admit a graceful labeling?

8/18/14 Edit: If anyone has a copy of a related reference, then I would be happy to see it. For now, I am accepting the answer below and considering the question answered in the affirmative: Yes. ...
2
votes
1answer
49 views

Graphs without nontrivial automorphism

I'm trying to solve two problems about graph automorphisms. I want to construct a bipartite graph without a nontrivial automorphism. I want to find the smallest possible number of nodes for a graph ...
0
votes
0answers
18 views

Cactus construction

Is there someone who can explain me how can one construct the cactus of the minimum cuts of a graph? Or someone who can suggest me a book about cactus construction theory? Thank you in advance
0
votes
0answers
27 views

Min cuts of a graph with odd edge-connectivity

Let $G=(V,E,w)$ be a weighted graph with integer weights, odd edge-connectivity $k(G)$ and $|E|$ minimum cuts that are linearly independent (i.e. every edge is contained in a minimum cut). Is it true ...
1
vote
1answer
26 views

Show that if the diameter of an undirected graph is $d$ then there exists some vertex separator $S\subseteq V$ of size $|S| \leq { n\over d-1} $

Show that if the diameter of an undirected graph is $d$ then there is some set $S\subseteq V$ with $|S| \leq \frac{n}{d-1} $ such that removing the vertices in S from the graph would break it into ...
2
votes
1answer
62 views

Help Needed Showing that $\chi(\overline{G \times H}) \leq \chi(\overline{G}) \times \chi(\overline{H})$

Where $\chi(G)$ denotes the chromatic number, $\overline{G}$ the graph complement, and $\times$ the Cartesian Graph Product: I need to show that $(\forall G,H)( \chi(\overline{G \times H}) \leq ...
1
vote
2answers
217 views

Hamilton Paths in n-Wheel Graph

According to wolfram, $n$-wheel graphs have $4(n-1)(n-2)$ Hamilton paths in them. $n$-wheel graph = http://mathworld.wolfram.com/WheelGraph.html http://mathworld.wolfram.com/HamiltonianPath.html ...
0
votes
0answers
26 views

Discrete Laplace operator (on graphs) - why are its units not the same as the continuous version of the Laplace operator?

From Wikipedia: Let $G = (V,E)$ be a graph with vertices $\scriptstyle V$ and edges $\scriptstyle E$. Let $\phi\colon V\to R$ be a function of the vertices taking values in a ring. Then, the discrete ...
0
votes
2answers
52 views

Chromatic recurrence

1) How do you prove the Chromatic recurrence theorem: $$χ(G;k)=χ(G−e;k)−χ(G·e;k)$$ I'm thinking by induction, but then you would have to assume something about the type of graph G...surely it can't ...
1
vote
1answer
59 views

Dijkstra's Algorithm- Two equal weights, one leads to a shorter path. What to do?

I am confused about this situation that happened to me as I was trying to solve a shortest path problem using Dijkstra's Algorithm. '$s$' is the starting point and '$t$' is finish. When I reach to ...
0
votes
1answer
23 views

Graph theory - Show that T has at least 2 vertices with deg(v)=1

So, i am going to show that, by using $$(T5): |E|=|V|-1$$ that the tree T has at least two vertices with degree 1. My attempts so far: We know that the sum of all degrees of a tree with n ...
1
vote
0answers
36 views

how to find a route in a graph

"Dr C is a tourist by nature, and wishes to visit each place once and return to her starting point. Dr D is an explorer, and wishes to traverse every road just once, in either direction; he is ...
1
vote
2answers
49 views

Proving a triangle with different edge colors exists in a graph.

This is again some homework translated (hopefully not too badly) from my book The graph $K_{n}$ is colored using $n$ different colors, in a way that each color is used at least once. Prove that there ...
0
votes
1answer
48 views

Discrete Math True or False

1.$\mathcal P(A\setminus B) = \mathcal P(A) \setminus P(B)$ These are power sets. False. 2.If $G$ is bipartite, then the complement of $G$ is disconnected. False. 3.Suppose $\mathcal F$ and ...
1
vote
1answer
68 views

Graph Theory Question On Exam Involving colorability of certain planar graph

I had a question on my exam and answered it using what I believe to be an Exhaustive Proof. The teacher marked it wrong, and while I understand there is a simple answer to the question, I would like ...
1
vote
2answers
52 views

Prove a 4-cycle exists in a graph with 100 vertices, each with degree of atleast 50

I hope I wrote the question well since it is my attempt at translating from the book. If it isnt clear enough, The question states that in every graph as described in the title a simple cycle of ...
1
vote
1answer
43 views

Proving breath first traversal on graphs [duplicate]

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
0
votes
0answers
42 views

Random looking Gray Codes or Hamiltonian Cycles on Hypercubes

Cyclic Gray codes come in many flavors and correspond 1-1 to Hamiltonian cycles on hypercubes. I would like to find a type that looks like a random walk on the hypercube. In a sense this is an ...
0
votes
1answer
54 views

Proofing a Reachable Node Algorithm for Graphs

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
0
votes
1answer
71 views

G as a graph without self loops and parallel edges with n vertices and m edges

EDITED to include c. Could someone help me understand this problem? I haven't been able to comprehend what I am supposed to do here. 1) Let G be a graph without self loops and parallel edges with n ...
2
votes
2answers
28 views

Construct Pairs of Non Isomorphic Graphs

I Have the following question : Give three examples of simple, connected graphs, all with 8 vertices with degrees 2, 2, 2, 2, 3, 3, 4 and 4, no pairs of which are isomorphic What is the best ...