# Tagged Questions

**3**

votes

**1**answer

75 views

### Extremal graph theory for directed graphs

In extremal graph theory, there are results such as
$$t(C_4,G)\geq t(K_2,G)^4,$$
where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...

**-1**

votes

**0**answers

24 views

### departure time/overlap algorithm [on hold]

i'm looking for "departure time/overlap algorithm" or any other idea.
Suppose you have n trains and each one has a performance profile(how much electricity they need at the current time while driving ...

**7**

votes

**4**answers

278 views

### Extremal examples for a folklore lemma on subgraphs of large minimum degree

It's a well known fact that a graph $G$ of average degree $d$ has a subgraph $G'$ of minimum degree at least $d/2$ and that the constant $1/2$ cannot be improved. The proof I know, which proceeds by ...

**1**

vote

**1**answer

122 views

### Degree Sequence Problem on $k$-Partite Graphs

The general Degree Sequence Problem asks for a simple undirected graph (that is a graph without self-loops and with no more than one edge between any pair of nodes) for which it holds that the degrees ...

**3**

votes

**2**answers

117 views

### Definition of the Moebius Ladder Graph

I found two different definitions of the Moebius Ladder Graph, whose essential difference is, whether the smallest one shall be $K_4$ or $K_{3,3}$.
according to Wikipedia ...

**2**

votes

**2**answers

111 views

### Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.
Now, if $G$ is a well-covered graph (where all maximal ...

**0**

votes

**0**answers

39 views

### Almost symmetric route in digraphs

Let $D=(V,E)$ be a digraph. A route of length $k$ in $D$ is a pair $L=(S,\sigma)$, where $S=(s_1,s_2,\dots,s_{k+1})$ is a sequence of $k+1$ elements of $V$, and ...

**14**

votes

**1**answer

1k views

### What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...)
Are there examples of results in "classical" [*] graph theory that have
been achieved by using simplicial ...

**1**

vote

**1**answer

97 views

### The edge chromatic number and pefectness of inflation of cubic graph

The inflation of graph $G$ is a graph $I(G)$
which is obtained by replacing each vertex $x$ by a complete graph
$K_{\deg(x)}$ and joining each edge to a different vertex of $K_{\deg(x)}$.
Let $G$ ...

**4**

votes

**1**answer

165 views

### Do graphs with large number of paths contain large chain minor?

Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
Question: Let ...

**1**

vote

**0**answers

102 views

### Knight's metric: ellipse and parabola

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...

**9**

votes

**3**answers

697 views

### Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...

**1**

vote

**1**answer

116 views

### Max order for which connected Cayley Graphs are known to be Hamiltonian

There is a well-known conjecture that all connected Cayley graphs are Hamiltonian.
For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?

**6**

votes

**1**answer

142 views

### Does high min degree and high odd girth imply near bipartiteness?

Say $G$ has odd girth at least $k$ and min degree $2n/k$. There is a classical result by Andrasfai, Erdos, and Sos that says that $G$ is bipartite. (Odd girth is the length of the shortest odd cycle ...

**4**

votes

**4**answers

308 views

### Determine if a graph has a large clique

This question is quite specific and practical. I hope it is still relevant for MO and will not be removed.
I have a collection $\mathcal{C}$ of graphs having from 5000-6000 vertices and edge density ...

**5**

votes

**1**answer

146 views

### The number of connected components of a generalized hypecube

For $n$ and $m$ positive integers $n>m\ge 1$ define a graph as follows.
The vertices are the binary strings of length $n$.
Two vertices are adjacent if they differ in exactly $m$ consecutive bits. ...

**2**

votes

**0**answers

32 views

### Balancing out edge multiplicites in a graph

Let $G$ be a multigraph with maximum edge multiplicity $t$ and minimum edge multiplicity $1$ (so that there is at least one 'ordinary' edge).
Is there some simple graph $H$ such that the $t$-fold ...

**5**

votes

**2**answers

171 views

### Average number of distinguished leaves in a binary tree

By a binary tree, I mean in this question a full rooted binary tree in which left and right child are labeled. A leaf of such a tree is a vertex of degree at most 1 (most references would probably ...

**0**

votes

**0**answers

74 views

### The largest size of a boolean subgraph (a hypercube) of a given graph

Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...

**3**

votes

**2**answers

145 views

### Is the domination number of a combinatorial design determined by the design parameters?

Let $D$ be a $(v,k,\lambda)$-design. By the domination number of $D$ I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$.
Is $\gamma(L(D))$ determined only by ...

**2**

votes

**0**answers

44 views

### Any one know the results of tensor decomposition by hypergraph partitioning?

Tucker Decomposition and CANDECOMP/PARAFAC (CP) Decomposition are two widely used tensor decomposition methods.
However, when we model the hypergraph into tensor, what's the connection between the ...

**3**

votes

**1**answer

112 views

### Is the domination number NP for non-bipartite graphs?

Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?

**2**

votes

**1**answer

108 views

### Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...

**3**

votes

**1**answer

219 views

### Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS. It doesn't appear at this time. I can't find any reference to it either. Has anyone seen it?
...

**6**

votes

**1**answer

149 views

### What is/are the best bound/s on the sum of squares of degrees in a graph?

Let $G$ be a graph with degrees $d_{1},\ldots,d_{n}$. I am interested in upper bounds on
$$
\sum_{i=1}^{n}{d_{i}^{2}}.
$$
An example is de Caen's bound:
$$
\sum_{i=1}^{n}{d_{i}^{2}} \leq ...

**7**

votes

**1**answer

392 views

### Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs

I believe correctness about clique in even-hole-free graphs
of graphclasses.org
and the paper Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas ...

**3**

votes

**1**answer

581 views

### Big binary tree as an induced subgraph

I believe this is true:
Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an
induced subgraph which is a subdivision of a large binary tree or the line ...

**10**

votes

**2**answers

171 views

### Big Mono-Chromatic Subgraphs of Vertex 2-Colourings

I'm not a graph theorist, but the following quantity came up in my work and I'm curious if it has been studied. Given a connected finite graph $\Gamma = (V,E)$ define: $$ c(\Gamma) = \min_{f : V ...

**0**

votes

**0**answers

48 views

### A particular method of removing edges from strong di-graphs

I have been mulling over a little puzzle I gave myself involving a particular type of iterative removal of edges from a digraph and I'm stuck -- thought I'd consult experts.
Start with an ...

**7**

votes

**1**answer

303 views

### A combinatorial problem concerned with logic circuits

Consider a logic circuit with two-bit gates only. The length of each gate is the number of bit lines that the gate crosses. How hard is to compute the maximum length for a given circuit? Notice that ...

**3**

votes

**2**answers

186 views

### Making a graph claw-free by adding as few edges as possible

Independent set is polynomial in claw-free graphs,
so I am wondering if this can approximate independent set.
By adding enough edges to $G$ and gets claw-free $G'$.
IS in $G'$ is IS in $G$, so this ...

**0**

votes

**0**answers

102 views

### What is the number of connected subgraphs with $n$ vertices of a labelled connected simple graph with $n$ vertices?

Suppose $G$ is a connected simple labeled graph. Let $n$, $e$, and $k$ be its number of vertices, edges, and the upper bound of the degree of a vertex, respectively. How many connected sub-graphs ...

**6**

votes

**1**answer

125 views

### Hamiltonicity criteria for sparse graphs

Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course).
There are three main classes of criteria for Hamiltonicity that I am aware of:
...

**1**

vote

**0**answers

46 views

### Perfect Matchings in Biclique Decompositions of Multigraphs

Suppose you have the $K_{2n}$ covered by a multigraph consisting of $2n-1$ bicliques, each consisting of a partition of the vertex set into two sets of equal size.
Here is a picture of $K_{6}$ with 5 ...

**3**

votes

**1**answer

50 views

### Are all (non-constant) symmetric submodular functions non-monotone?

I am trying to show (if possible) that symmetric submodular functions are non-monotone (excluding constant sub-modular functions).
Recall that a submodular function $f : 2^{\Omega} \rightarrow R$ is ...

**4**

votes

**2**answers

136 views

### class 1 vs class 2 in regular graphs

Vizing's theorem states that a graph can be edge-colored in either $\Delta$ or $\Delta+1$ colors, where $\Delta$ is the maximum degree of the graph.
A graph with edge chromatic number equal to ...

**2**

votes

**3**answers

204 views

### Regular graphs whose neighbourhoods induce matchings

Studying some problem I've arrived to the following notion.
Let a $2r$-regular graph $G$ be called neighbour-matching if $N(v) = rK_2.$ In other words, the neighbourhood of any vertex induces a ...

**1**

vote

**1**answer

116 views

### Upper-bound for maximal-cliques on perfect graphs

It has been proved by Moon and Moser in 1965 that any finite simple graph has at most $3^{|V|/3}$ maximal cliques. Still, some hereditary classes of graphs have very few maximal cliques in comparison ...

**0**

votes

**0**answers

65 views

### Existence of a sequence of (almost) Moore irregular graphs embedded on closed surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. I'm interested in disproving the existence of a certain configuration of simple closed curves on $S_{g}$. I'd be happy to go into more ...

**3**

votes

**1**answer

169 views

### Equivalence of Hadamard Graph and Hadamard Matrix

I'm reading Distance Regular Graphs by Brouwer, Cohen, and Neumaier. In section 1.8, they explained Hadamard graphs.
Conversion from a Hadamard Matrix into a Hadamard Graph
An $n$-Hadamard graph $G$ ...

**1**

vote

**1**answer

171 views

### Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...

**15**

votes

**0**answers

206 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 $H$.
...

**3**

votes

**0**answers

70 views

### What is this expander-mixing-type graph property?

Fix $C>0$. I am interested in graphs with the following mixing property:
$$\Big|E(S,T)-\frac{1}{2}|S||T|\Big|\leq C\sqrt{|S||T|\max\{|S|,|T|\}}$$
for every disjoint $S,T\subseteq V$. Note that ...

**2**

votes

**1**answer

143 views

### Regular graphs with $a$ and $b$ Hamiltonian edges

Special case of this question.
Let $G$ be $r$-regular Hamiltonian graph.
An $a$ edge is an edge which is on every Hamiltonian cycle.
A $b$ edge is an edge which is on no Hamiltonian cycle.
$a(G)$ ...

**5**

votes

**1**answer

215 views

### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

**7**

votes

**2**answers

289 views

### Graphs with many edges avoided by Hamiltonian cycles

Let $G$ be a $3$-connected Hamiltonian graph with at least one edge that belongs to each H-cycle of $G$. Some authors (e.g. in the link given here) call such an edge an a-edge and an edge that belongs ...

**4**

votes

**2**answers

177 views

### Number of unlabelled planar graphs

What are the best known bounds on the number of non-isomorphic (unlabelled) planar graphs on $n$ vertices? Is there a simple proof that this number is at most exponential in $n$?

**1**

vote

**1**answer

196 views

### Is this Graph parameter known?

Let $\lambda(G)$ denote the edge-connectivity of $G$.
Consider the following parameter:
$\rho(G) = \max_{X \subset V(G)} \min(\lambda(G[X]), \lambda(G[V(G) - X]))$
Has this parameter been studied? ...

**5**

votes

**0**answers

106 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**2**

votes

**1**answer

146 views

### About an equivalent to Tutte's 5-flow Conjecture

A while back I remember reading that F. Jaeger proved that Tutte's $5$-flow conjecture is equivalent to a statement about the co-planarity of a certain set of points in some euclidean space. But I ...