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, ...

learn more… | top users | synonyms

32
votes
0answers
999 views

Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...
28
votes
0answers
685 views

3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$. Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...
27
votes
0answers
470 views

What does this connection between Chebyshev, Ramanujan, Ihara and Riemann mean?

It all started with Chris' answer saying returning paths on cubic graphs without backtracking can be expressed by the following recursion relation: $$p_{r+1}(a) = ap_r(a)-2p_{r-1}(a)$$ $a$ is an ...
21
votes
0answers
735 views

Non-linear expanders?

Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
19
votes
0answers
306 views

Zero curves of Tutte Polynomials?

There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...
19
votes
0answers
298 views

Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$. ...
18
votes
0answers
546 views

Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...
18
votes
0answers
373 views

Planar minor graphs

The theorem of Robertson-Seymour about graph minors says that there exists no infinite family of graphs such that none of them is a minor of another one. Apparently, it came as a generalization of ...
16
votes
0answers
262 views

Simpler proofs of certain Ramsey numbers

The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture. But for bigger Ramsey ...
15
votes
0answers
311 views

Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction. Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
15
votes
0answers
855 views

Is every k-edge-connected graph also k-trail-ordered?

This is an old question of Aradhana Narula-Tam and Philip Lin that I think deserves wider circulation. It appeared in Discrete Math. 257 (2002), page 613, but not many people have looked at it and it ...
15
votes
0answers
783 views

Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X. For a real number p between zero and one, we consider a ...
14
votes
0answers
304 views

Monotone embedding of complete binary tree in hypercube

Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to ...
14
votes
0answers
642 views

A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed spanning trees. Suppose that There no subset X of vertices so that there is precisely one directed edge from X ...
13
votes
0answers
506 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in ...
13
votes
0answers
359 views

How much must deleting a spanning tree reduce edge-connectivity?

Suppose you have a 100-edge connected graph (e.g. an infrastructure network). You want to delete the edges of a spanning tree, any spanning tree you choose (e.g. to sell a connected subnetwork). What ...
12
votes
0answers
288 views

Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of ...
12
votes
0answers
1k views

Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...
11
votes
0answers
263 views

The Universal Labeling of graph

The universal labeling of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ ...
11
votes
0answers
1k views

A new lower bound for the chromatic number of a graph?

Let $S_{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph $G$. Let $S_{-}(G)$ denote the sum of the squares of the negative eigenvalues and $q$ the ...
11
votes
0answers
730 views

Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...
11
votes
0answers
375 views

Reconstruction Conjecture and Partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant ...
11
votes
0answers
300 views

Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
10
votes
0answers
347 views

Connections between Riemann hypothesis for curves over finite fields and Ramanujan property for graphs

this question relates to the beautiful construction of expander graphs using Cayley graphs of $PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, Elementary Number Theory, ...
10
votes
0answers
309 views

Spectral theory of graph Laplacian besides $\lambda_2$

Most of what I've seen about the spectral theory of the graph Laplacian concentrates on $\lambda_2$, the second-smallest eigenvalue. This eigenvalue contains information regarding the connectivity of ...
9
votes
0answers
99 views

Cycles of length $2^n - 2$ in the De Bruijn graph

It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$. ...
9
votes
0answers
292 views

Is there an “Erlangen Program” for Graph Theory?

There are certain graph theoretic problems (especially optimization problems), whose solution-subgraph (i.e. the set of vertices and edges)), is invariant under certain modifications (especially ...
9
votes
0answers
558 views

Shortest path in Cayley graphs

The standard way to find the shortest path between 2 vertices, $v_1$ and $v_2$, of an undirected graph is BFS (breadth first search) which takes time $O(|E|)$ and space $O(|V|)$ (where $E$ is the set ...
9
votes
0answers
373 views

Has the technique of “sprinkling” been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
8
votes
0answers
130 views

Edge-colorings of plane graphs: do you know references where the following questions are studied?

Let $G$ be a plane graph (or more generally, a graph embedded on a surface) with a proper edge-coloring of $G$ with $k$ colors $\{1,\ldots,k\}$. I am interested in studying the cyclic permutations of ...
8
votes
0answers
188 views

How many n/2-cycles can a cubic graph have

Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have? A random cubic graph has ...
8
votes
0answers
263 views

Colouring a graph whose edge set is a special union of cliques

I am trying to show that a certain family of graphs can always be properly coloured with at most $6$ colours (where "properly coloured" means that each vertex gets a colour and no edge has both ends ...
8
votes
0answers
362 views

A counterexample to a conjecture of Nash-Williams about hamiltonicity of digraphs?

Maybe I am missing something, but found potential counterexample to a conjecture of Nash-Williams. According to HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS The outdegree and indegree sequences of ...
8
votes
0answers
405 views

One more coloring question

This question is related to my previous questions, say, this one and this one. Let $G$ be an infinite graph of bounded degree, and $\lambda>0$. Let $k=k_G(\lambda)$ be the minimal number of colors ...
8
votes
0answers
405 views

Abelian sandpile models

This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as ...
8
votes
0answers
366 views

An extremal problem for graphs having every edge contained in a 4-clique

This is a follow-up to Graphs with many triangles but few complete graphs on 4 vertices I'm looking for an upper bound for the difference between the number of edges and the number of 4-cliques in a ...
8
votes
0answers
514 views

Is the dominating set problem restricted to planar bipartite graphs of maximum degree 3 NP-complete?

Does anyone know about an NP-completeness result for the DOMINATING SET problem in graphs, restricted to the class of planar bipartite graphs of maximum degree 3? I know it is NP-complete for the ...
8
votes
0answers
116 views

Disjoint Rooted Paths with Specified Patterns

Let $S:=$ { $s_i : i \in [k]$ } and $T:=$ { $t_i : i \in [k]$ } be disjoint subsets of vertices of a graph $G$. Furthermore, let $A$ be a subset of $S_k$ (the symmetric group on $[k]$). A set of ...
7
votes
0answers
70 views

A separation property of graphs of bounded tree-width

The following separation property of trees is well-known and in fact easy to prove (see e.g. the paper "Covering a hypergraph of subgraphs" by Noga Alon, Lemma 2.2) Let $T$ be a tree and $r, m$ ...
7
votes
0answers
65 views

Are graphs with sparse $r$-balls necessarily sparse?

Let $G$ be an unweighted undirected graph with the following property: For some integer $r$, for all nodes $v$, we have $$\frac{\sum \limits_{u \in B(v, r)} \deg(u)}{|B(v, r)|} \le D$$ where $B(v, r) ...
7
votes
0answers
125 views

De Bruijn sequence inside De Bruijn sequence

A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1a_2\ldots a_{2^n}$, with $a_i∈\{0,1\}$, and such that each of the $2^n$ binary $n$-tuples occurs exactly once in $S$. What is ...
7
votes
0answers
286 views

Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
7
votes
0answers
81 views

Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
7
votes
0answers
124 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 ...
7
votes
0answers
238 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 ...
7
votes
0answers
180 views

Genus of the graph $K_{4,2,2,2}$

I have ask this question in math.stackexchange, here. Since, there is no answer and apart from that i feel that the problem is difficult, i would like to ask it here. The problem is to find the genus ...
7
votes
0answers
128 views

A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$

Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices (A138464 on the OEIS). The first few values of $F(n;i) \pmod n$ are listed below: $$\begin{array}{r|rrrrrrrrrrr} & i=0 ...
7
votes
0answers
202 views

Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
7
votes
0answers
152 views

How quickly can we test if a graph is distance-regular?

A (simple, finite, connected) graph $G$ is distance regular if there exist integers $b_i,c_i,i=0,...,D$ such that for any two vertices $x,y$ in $G$ and distance $i=d(x,y)$, there are exactly $c_i$ ...
7
votes
0answers
204 views

Is there an analog of Khovanov homology for edge deletion-contraction-extraction?

Motivated by Khovanov's categorification of the Jones polynomial, several authors have worked on the categorification of graph invariants. For the chromatic polynomial some references are: "A ...