**-4**

votes

**0**answers

59 views

### what's wrong in the attack for 4CT? [on hold]

Dirac proved that :
Every 4-chromatic graph contains a K4-subdivision. --- Good
in the paper : A property of 4-chromatic graphs and some remarks on critical graphs J. London Math. Soc., 27 (1952), ...

**2**

votes

**0**answers

56 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

55 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

80 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 ...

**0**

votes

**0**answers

162 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

102 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

283 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

104 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

369 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

126 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 ...

**1**

vote

**1**answer

76 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

91 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

118 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

61 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 ...

**5**

votes

**1**answer

436 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

60 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

49 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 $\chi_e(H)$ ...

**-1**

votes

**1**answer

106 views

### Reducing chromatic number

(1) Is there an estimate for maximum number of edges in a $k$ colorable $v$ vertex $d$ degree graph with genus $g$? Call this $|E|$?
(2a) What is a good estimate for worst case number of edges that ...

**2**

votes

**1**answer

213 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

73 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

58 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

74 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

81 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

90 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

105 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

62 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 ...

**1**

vote

**1**answer

134 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 ...

**12**

votes

**1**answer

165 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

72 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

51 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

121 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

367 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

279 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

283 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

234 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

271 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

234 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

136 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

78 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 ...

**6**

votes

**3**answers

150 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

77 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 ...

**2**

votes

**2**answers

60 views

### Minimal edge color on constraints

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every even simple cycle contains an odd number ($>1$) of colors much larger than $(\log n)^\beta$ or ...

**2**

votes

**1**answer

98 views

### Images of interval edge coloring

I found out the definition of interval edge colorings, concept put by Kamalian in various papers but could not find a graph depicting an example. Where can I find pictures of explicit examples of ...

**4**

votes

**0**answers

67 views

### Existence of certain graph gadget related to coloring odd hole free graph

Wondering about the existence of graph gadget related to coloring
(or 3-coloring) odd hole free graphs.
Let $G$ be simple $k$-chromatic connected graph with two
vertices $u,v$.
Is it possible $G$ to ...

**0**

votes

**0**answers

27 views

### Complexity of edge coloring graphs of sufficiently large maximum degree

I am interested in the complexity of edge coloring
graphs with $\Delta(G) > |V(G)|/3$.
This is closely related to the Overfull conjecture (OC).
Conjecture/Question: If a simple graph G with n ...