**2**

votes

**0**answers

39 views

### A generalization of the definition of edge coloring and a related problem

A generalization of the definition of edge coloring:
For any positive integer $k\geq 2$, a $k$-path edge coloring of a graph $G$ is a assignment of colors to the edges of $G$ so that for every path $...

**0**

votes

**0**answers

22 views

### Permutation Invariant Color Class

$G$ is a $d$ regular graph, it has $n$ vertices.
$S_n$ acts on $n$ vertices of graph $G$.
Question: Does there exist a coloring algorithm for which color classes is invariant under all ...

**0**

votes

**0**answers

23 views

### Example/ Explanation of Weisfeiler-Lehman method

I am trying to read An Optimal Lower Bound on the Number of
Variables for Graph Identification. On page 3 (4th paragraph), it is written-
It might color vertices and edges implicitly by using $k$-...

**0**

votes

**0**answers

26 views

### Possible Number of Repetation of a Submatrix

Notation:
$H$ is the adjacency matrix of graph $H'$ respectively. $H_k$ is the block or sub-matrix of matrix $H$. The adjacency matrix of graph $H_k \cup H_e$ (subgraphs of $H'$) is $M_{(k,e)}$ ...

**0**

votes

**0**answers

137 views

### Graph Coloring: Two adjacent vertices share same color

Consider, subgraphs $G_1, G_2,...... G_x$ of graph $G$. Each subgraph has $k$ vertices.
Now, Fix subgraph $G_1$ and consider another subgraph $G_k$ where $1 <k \le x$.
The edge set ...

**3**

votes

**0**answers

65 views

### Is the chromatic number of the graph of the permutahedron known?

The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the ...

**2**

votes

**0**answers

61 views

### Number of $(2n-1)$-edge-colorings of the complete graph $K_{2n}$

I just started reading about graph theory and have a question (which might be trivial). How many $(2n-1)$ edge colorings of $K_{2n}$ are there?
A vaguer question: can I write $K_{4n}= K_4 + K_4 +.......

**0**

votes

**1**answer

73 views

### Choosing directed subgraph in a triangulation

Consider triangulation $T.$
Is it always possible to choose such a subgraph $H$ of $T$ that has a common edge with every face of $T$ and can be directed in such way that indegrees of all vertices of ...

**4**

votes

**1**answer

81 views

### Choice number of embedded graphs

For given $g$, consider the family of graphs which may be embedded to the compact orientable surface of genus $g$. For this family, consider maximal clique $\alpha(g)$, maximal chromatic number $\chi(...

**0**

votes

**0**answers

165 views

### SDP based heuristics for graph coloring

This is a question about semidefinite programming heuristics for graph vertex coloring based on the Lovasz theta number such as "Approximate Graph Coloring" by Karger, Motwani and Sudan or "A ...

**3**

votes

**2**answers

105 views

### Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...

**7**

votes

**1**answer

286 views

### Choosing two-colorable subgraph in a triangulation

Consider a planar graph $G$ which is a triangulation.
Is it possible to find a two-colorable subgraph $H$ of $G$ which has a common edge with every face of $G$?
It is known that it is not always ...

**-1**

votes

**1**answer

106 views

### Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem

In the setting of the Hadwiger-Nelson problem, two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and ...

**13**

votes

**2**answers

374 views

### Has there been a computer search for a 5-chromatic unit distance graph?

The existence of a 4-chromatic unit distance graph (e.g., the Moser spindle) establishes a lower bound of 4 for the chromatic number of the plane (see the Nelson-Hadwiger problem).
Obviously, it ...

**5**

votes

**2**answers

131 views

### Finite graph colorings without symmetries

Let $G$ be a connected finite simple graph with vertex set $V$, $F$ a finite set and let $\Delta(G)$ denote the degree of $G$, i.e. $\Delta(G)= \max_{v\in V} \deg(v)$. We say that a coloring $\phi\...

**1**

vote

**1**answer

78 views

### many 5-list colorings

If a graph is 4-list-colorable, then it is easy to see that it has exponentially many 5-list colorings.
This is the first sentence in [Exponentially many 5-list-colorings of planar graphs,JCTB,97 (...

**5**

votes

**2**answers

92 views

### Majority coloring for directed graphs

I came up with the following coloring concept when studying neural networks (which are often modelled using directed graphs). No idea whether there is already an established name for it.
If $X$ is a ...

**8**

votes

**1**answer

124 views

### How many uniquely colored degree two vertices in 3-coloring of subcubic graph?

Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph?
I'm interested in any similar results as ...

**1**

vote

**0**answers

65 views

### A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...

**1**

vote

**0**answers

114 views

### 2-edge colorable graph approximation

A 2 edge-colorable graph is a graph in which we can color the edges with two colors, in a way such that no edges of the same color share a vertex.
Given a graph G = (V,E) I want to find a 2 edge-...

**5**

votes

**1**answer

439 views

### Is this graph 3-colorable?

Consider the permutations of $0,1,1,2,2,3,3.$ Each permutation is corresponding to a vertex in graph $G$. So, the graph $G$ has $630$ vertices.
Each vertex has exactly 6 neighbors. $P$ is connected $...

**2**

votes

**0**answers

70 views

### Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...

**2**

votes

**1**answer

50 views

### Bounds on chromatic index

Let $H$ be a hypergraph of maximum vertex-degree $\Delta$. (That is, for all vertices $x$, we have $| \{ e \in H \mid x \in e \} | \leq \Delta$) Are there any bounds on the chromatic index $\chi_e(H)$ ...

**2**

votes

**1**answer

216 views

### Proof that it's possible to colour all elements in set, that all subsets will be bicolored

(For my easy understanding, let me rewrite the question. The author should feel free to remove my edit or... accept it; I am leaving the original formulation at the end intact).
=================
...

**5**

votes

**0**answers

74 views

### Chromatic numbers for coloring-constrained graphs

I am interested in any and all articles about chromatic numbers applying to constrained colorings of a graph. For example, if a graph must be (properly) colored so that there is a 2-color path ...

**2**

votes

**0**answers

59 views

### Planar triangulations for which all distinct 4-colorings consist of exactly 6 Kempe chains

Are there any internally 6-connected planar triangulations other than the icosahedron all of whose distinct 4-colorings consist of exactly 6 Kempe chains, one for each of the 6 color-pairs?
Addendum: ...

**4**

votes

**1**answer

76 views

### Show existence maximal clique of order $s$ in an multigraph where each vertex is colored with a set of colors

You are given a multigraph $G$ with $n$ vertices as follows:
$V := (v_1, v_2, \dots ,v_n)$
$C := \{c_1, c_2, \dots\}$, be an infinite set of colors.
$f: V \rightarrow \mathbb{P}_{\le m}(C) $, a ...

**1**

vote

**0**answers

33 views

### Colorful Neighborhoods

Given:
$G:=\{V= \{v_1,\ ...\ v_n\},E\subset V\times V\}, n<\infty$, a symmetric complete, simple graph
$w:=\ \ E \ni e_{ij}\mapsto \mathbb{R}^+$, a weight function for the edges of $G$
$K:=\{c_1,\ ....

**2**

votes

**1**answer

82 views

### Partitioning the vertex set of a planar bipartite graph into a tree and an independent set

Let $G = (V, E)$ be a planar bipartite graph such that there is a partition $(V1, V2)$ of $V$ where $V1$ induces a tree and $V2$ induces an independent set.
Is there a characterization of such ...

**0**

votes

**1**answer

100 views

### Maximal chromatic number with a fixed number of edges

Given a graph $G$ with $m$ edges, what is the maximum chromatic number $\chi(G)$ that the graph can have?
My guess is that $\chi(G) \leq r(m)$ where $r(m) := \max\{k\in \mathbb{N}:
\frac{k(k-1)}{2} \...

**-1**

votes

**2**answers

106 views

### Do graphs with $\omega(G) = \chi(G)$ grow “common” as $|V|$ grows large?

On the set $[n]:= \{1,\ldots,n\}$ we consider the set $${\cal P}_2([n]) = \big\{\{a,b\}: a,b \in [n], a\neq b\big\}.$$
Since $$|{\cal P}_2([n])| =2^{n \choose 2}$$ there are exactly $2^{n\choose 2}$ ...

**0**

votes

**1**answer

63 views

### Weak Erdos graphs

We call a finite, simple, undirected graph $G=(V,E)$ an $n$-Erdos graph if there are $n$ subsets $S_1,\ldots, S_n$ of $V$ such that
$V = \bigcup_{n=1}^n S_n$;
each $S_k$ has $n$ elements for $k\in\{...

**1**

vote

**1**answer

136 views

### Node-edge coloring of graphs

There must be work on this concept, but I am not finding it through
searches, perhaps using the wrong terminology.
Define a node-edge coloring of a graph $G=(V,E)...

**12**

votes

**1**answer

167 views

### Graphs with a coloring that majorizes all other colorings

By a coloring of a graph $G = (V,E)$ I mean a map $\kappa:V\to\mathbb{N}$ such that $\kappa(u)\ne \kappa(v)$ whenever $u$ and $v$ are adjacent. (Sometimes this is called a proper coloring but I am ...

**1**

vote

**0**answers

23 views

### Performance guarantee of RLF [closed]

I cannot manage to find the performance guarantee of the Recursive Largest First (RLF) algorithm for approximating the chromatic number of a graph.
I know DSATUR has a $\mathcal{O}(n)$ guarantee, ...

**4**

votes

**0**answers

74 views

### A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...

**2**

votes

**0**answers

52 views

### The numbers of edge colorings and partitions into vertex covers

Let $G = \langle X \cup Y, E\rangle$ be a $d$-regular bipartite graph such that $|U|=|V|=n$, and assume that $d$ divides $n$. Let $F(G)$ denote the number of proper edge colorings of $G$ using $d$ ...

**-1**

votes

**1**answer

122 views

### Graph such that edge contraction increases chromatic number

Let $G=(V,E)$ be a simple, undirected graph with the following properties:
Contracting any edge increases the chromatic number by $1$;
For each minor $M$ of $G$ we have $\chi(M) \leq \chi(G) + 1$.
...

**6**

votes

**1**answer

375 views

### Graphs in which any two odd cycle have a common vertex

Let $G$ be a graph in which any two odd cycles have a common vertex. It is easy to see that $\chi(G)\leq 5$ (choose minimal odd cycle $C$, use two colors for $G\setminus C$ and three colors for $C$). ...

**7**

votes

**1**answer

281 views

### Chromatic numbers of nowhere dense graphs

Given $c\in (0,1)$ and a graph $G=(V,E)$ such that any subset $U\subset V$ contains an independent subset of cardinality at least $c|U|$. Does it allow to bound the chromatic number $\chi(G)$ by the ...

**16**

votes

**0**answers

287 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 ...

**11**

votes

**1**answer

235 views

### What is the chromatic number of the “conic hypergraph” on a non-singular plane cubic?

Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 distinct points of the same color?
If so, what is the smallest ...

**8**

votes

**0**answers

134 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 ...

**7**

votes

**4**answers

274 views

### Graph homomorphisms and line graph

If $G,H$ are simple, undirected graphs, we say that they are in a hom(omorphism)-relation if there is a graph homomorphism from $G$ to $H$ or from $H$ to $G$.
For any graph $G$ let $L(G)$ denote its ...

**5**

votes

**2**answers

238 views

### How can I prove that these two graph coloring problems are polynomial time equivalent?

Given a graph $G(V,E)$. The standard $k$-coloring problem consists in finding a feasible coloring (no two adjacent nodes share the same color) of the nodes with $k$ colors. Let this problem be $P_1$.
...

**3**

votes

**3**answers

145 views

### Algorithm to determine isomorphism of 2 maximal planar graphs

I read on wikipedia that there are efficient algorithms to answer the question whether 2 (maximal) planar graphs F and G are isomorphic. However, after some (IMHO) substantial searching I don't seem ...

**5**

votes

**1**answer

82 views

### Algorithm for 2-coloring classes of 3-uniform hypergraphs

Hi all and thanks in advance for your efforts.
I'm interested in 2-coloring 3-uniform hypergraphs. I know that in general, the problem of deciding if a 3-uniform hypergraph is 2-colorable is NP-hard....

**7**

votes

**3**answers

153 views

### Colorings of triangulations as a generalization of the four-color problem

My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two ...

**2**

votes

**0**answers

79 views

### Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...

**0**

votes

**0**answers

83 views

### A constrained minimum edge coloring

Is minimum number of colors needed to color edges of complete graph $K_n$ so that every even simple cycle contains at least one color assigned to odd number of edges at most $\beta n$ where $\beta\...