Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, ...

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2
votes
0answers
47 views

When polynomial GI implies polynomial (edge) colored GI?

(edge) colored graph isomorphism is GI which preserves the colors (of edges if it is edge colored). There are several reductions using transformations/gadgets of (edge) colored GI to GI. For edge ...
2
votes
0answers
47 views

Looking for similar centrality measurement on graph

I'm working on a graph problem somehow related to centrality measurement. Given an undirected, unweighted tree $T$ and a vertex $v$, let $D_i(v)$ be the set of vertices in $T$ that are i hops from ...
2
votes
0answers
61 views

Properties of a smallest tournament with domination number $k$

For some tournament $T$, let $\gamma(T)$ denote the cardinality of a smallest dominating set of $T$. Denote by $f(k)$ the minimum number of vertices of a tournament $T$ having $\gamma(T) = k$. From ...
1
vote
1answer
66 views

Two definitions of genus for circle graphs

In the (very nice) article of Goldstein and Turner untitled Applications of Topological Graph Theory to Group Theory, the following definitions can be found: Definitions: A circle graph is a pair ...
6
votes
2answers
203 views

Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...
6
votes
1answer
148 views

Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...
2
votes
0answers
116 views

NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC. http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...
0
votes
1answer
41 views

Resource on characterizations or properties of traceable graphs

I am looking for some resources that provide information on traceable graphs(paths containing a hamiltonian path). I have found a lot of information on hamiltonian graphs, but none on traceable ...
0
votes
0answers
71 views

asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question. Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf Namely theorem 5. Now, feel ...
5
votes
1answer
56 views

Is a connected graph uniquely determined by its weighted 2-step graph?

This is an extension of a previous question: Is a graph uniquely determined by its weighted 2-step graph?. In that question I asked about arbitrary graphs; in this question I restrict to connected ...
11
votes
0answers
381 views

Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...
0
votes
0answers
57 views

How many edges are in the kth-iterated common neighborhood knight's graph?

For terminology see https://www.academia.edu/2180382/The_Common_Neighborhood_Graph_and_Its_Energy) It has been known for quite some time by players that a vertex whose corresponding square is in the ...
6
votes
0answers
71 views

Set system with prescribed intersection sizes

Questions: What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$? In ...
1
vote
1answer
98 views

edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...
7
votes
2answers
266 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...
5
votes
1answer
130 views

Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define: $G$ has property A iff it is edge-transitive. $G$ has property B iff each edge belongs to the same number of ...
2
votes
0answers
163 views

Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate polynomial which counts the number of colorings of $G$ with $x$ colors for natural $x$. Graph is not $k$ colorable iff $P(G,k)=0$. The ...
1
vote
1answer
125 views

Vertex transitive and edge transitive and line graph

How can we find the proof of the following statement: An undirected graph is edge transitive if and only if its line graph is vertex transitive.
24
votes
3answers
591 views

Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...
10
votes
2answers
485 views

Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$, there exists some graph $G$ whose every embedding into $\mathbb{R}^3$ (or into $\mathbb{S}^3$) contains a cycle that realizes $K$? I know the ...
2
votes
1answer
212 views

How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem). Restricted GA Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...
0
votes
0answers
41 views

Complexity of graph isomorphism in $(P_4 \cup K_1,\overline{3K_2})$-free graphs

Related to this question where isomorphism preserving transformation maps triangle-free graphs to $(P_4 \cup K_1,\overline{3K_2})$-free graphs. What is the complexity of graph isomorphism in $(P_4 ...
3
votes
1answer
104 views

Graph transformation related to graph isomorphism

Basically got graph transformation related to graph isomorphism. Define $G \to G'$. $V(G')=V(G) \cup E(G)=\{v_1\ldots v_n\} \cup \{e_1\ldots e_m\}$. Call $v_i$ vertices $v'$ and $e_i$ vertices $e'$. ...
4
votes
1answer
505 views

Surprising connection between linear algebra and graph theory

What is the intuition for linear algebra being such an effective tool to resolve questions regarding graphs? For example, one can determine if a given graph is connected by computing its Laplacian ...
0
votes
0answers
26 views

Splitting lemma for digraph and preserving local rooted-edge connectivity?

Let $G$ be a directed graph. $\lambda(x,y,G)$ is the maximum number of edge disjoint paths from $x$ to $y$ in $G$ The local $r$-rooted connectivity of $x$ in $G$ is $\lambda(r,x,G)$. The global ...
2
votes
1answer
128 views

Minimum length path touching $n$ circles

Given $n$ non-overlapping circles of radius $1$ (i.e., the distance between the centers of any two circles is greater than $2$), how to find the minimum length path (the path can be of any form) that ...
3
votes
0answers
41 views

Algorithm to construct metric space endomorphism with controlled square

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be ...
0
votes
1answer
72 views

N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...
2
votes
1answer
83 views

About the diameter of a graph after removing orientation

This question was posted a few days ago on the Mathematics StackExchange, but so far it has not been answered. Let $G$ be a strongly connected directed graph of diameter $D$, and suppose that we ...
4
votes
0answers
35 views

What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even when restricted to ...
1
vote
2answers
176 views

Strongly connected DAG from any connected undirected graph?

I have the following question. It seems likely to be true - can anyone provide a standard reference? Given: A connected, undirected graph. Question 1: Can we assume a single direction for each edge ...
7
votes
0answers
124 views

Algorithms for computing the Resilience of Graphs

The definition of resilience with a graph $G$ w.r.t to a monotone property $\mathcal{P}$ is well known. (Global resilience) Let $\mathcal{P}$ be an increasing monotone property. The global ...
3
votes
0answers
84 views

Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large. An agent can move in the area at ...
0
votes
0answers
97 views

Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$

Valiant shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$ to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$ for certain efficiently computable $t(F)$ and matrix ...
6
votes
2answers
174 views

Find multiple non-adjacent paths in a graph

Consider a non-directed graph. I want to find as many non-adjacent paths as possible from a source $s$ to a destination $t$. Two paths $P_1$ and $P_2$ are said to be non-adjacent to each other if none ...
4
votes
1answer
114 views

Product of geodesic distances

I'm working on trying to show this, but can't seem to get started. No guarantees that it is true, but other conditions on the adjacency matrix that make it true or a counter example are helpful. ...
0
votes
2answers
80 views

different way of selecting a random graph

Consider having a 'base' graph $G=(V,E)$ and selecting each vertex with independent probability $p$ and having the induced subgraph of $G$ with all 'selected' points as your random graph. Has this ...
0
votes
4answers
111 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 ...
3
votes
1answer
105 views

Graphs of lines on del Pezzo surfaces

Let $k$ be an algebraically closed field. To any del Pezzo surface $S$ over $k$ we may associate its graph of lines, which has one vertex for each line and an edge (with multiplicity if required) ...
2
votes
1answer
155 views

Clique problem for regular graphs

I am looking for NP complete results for cliques in regular graphs. For example is the general problem of determining if a regular graph on n vertices has an n/2 clique NP-complete? (obviously the ...
0
votes
0answers
43 views

Paths on Cartesian products of graphs satisfying linear constraints

Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ ...
1
vote
0answers
72 views

Empty node in cactus construction

Is there a necessary condition for not having empty node in the construction of the cactus of the minimum cuts of a graph? In particular is there a necessary condition for not having empty k-junction ...
0
votes
0answers
45 views

Correlation between attributes in a binary graph

Given an unrooted binary tree whose leaves are vertices of degree one that are labelled bijectively by a set $S$. We define a categorical attribute $A$ ($|A|<<|S|$) and each leaf is assigned a ...
1
vote
0answers
107 views

Find a path that covers as many nodes as possible

I have the following interesting problem. Given a graph $G$, an agent starts to mark nodes in $G$ in the following way: it marks all nodes within distance $d$ from it. Now the question is to find the ...
4
votes
1answer
112 views

Expected number of connected components in a random graph

For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value? I'm specially interested in what happens for small values of p, ...
-4
votes
1answer
151 views

Bipartite graph [closed]

First of all, thank you for your time to reading my post. I am a researcher but not a mathematician, i have some difficulties in solving a math problem, that why i am here to ask your help. I just ...
0
votes
1answer
155 views

Kneser graphs eigenvalues

Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...
6
votes
2answers
199 views

Conjecture: for perfect graphs the fractional chromatic index rounded up equals the chromatic index

Let $\chi'_f(G)$ be the fractional chromatic index. Based on limited experiments (up to 8 vertices and few larger graphs), I suspect: Conjecture For perfect graphs $\lceil \chi'_f(G) \rceil = ...
2
votes
1answer
93 views

Estimate for the travelling salesman problem for balls inside a grid

This question is probably easy but I only have "tedious case checking" proof strategy in sight, and I'm sure there should be a reference lying around... The question concerns the TSP problem (with ...
1
vote
0answers
99 views

random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...