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6
votes
0answers
93 views

What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...
0
votes
2answers
69 views

Coloring of a normal map

can the following proposition be proved? If so please suggest a method. Can Kempe’s Argument be used for proof ? Proposition: A normal map has a colouring of countries by 4 colours iff the edges of ...
8
votes
1answer
307 views

Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...
0
votes
2answers
42 views

Coloring maximal independent sets with 1 color

Let $G$ be a graph and $M\subseteq V(G)$ be a maximal independent set. Is there a coloring $c:V(G)\to\chi(G)$ such that $c$ is constant on $M$? (The answer is positive for graphs with infinite ...
1
vote
2answers
70 views

Regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $c k$ regular graph on $v$ vertices that can be given an edge coloring with $c$ colors, such that the ...
1
vote
1answer
54 views

Coloring algorithm maximising color difference between neighbors

Consider a graph and a set of ordered colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two ...
0
votes
1answer
170 views

Coin graph is 4-colorable

How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.
2
votes
1answer
188 views

On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections

Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements. We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...
2
votes
1answer
66 views

Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...
1
vote
0answers
104 views

Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com: http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using ...
5
votes
2answers
249 views

Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$). For odd integers $n$ we have $n=\chi(K_n) = ...
0
votes
0answers
74 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 ...
2
votes
0answers
167 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 ...
6
votes
2answers
201 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 = ...
8
votes
1answer
327 views

Postnikov's approach to perfect matchings of graphs

Over a decade ago Alexander Postnikov developed his own way of looking at perfect matchings of bipartite plane graphs. As I recall, he starts with a 2-coloring of the square grid and creates a new ...
6
votes
0answers
136 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 $v$ ...
34
votes
6answers
3k views

Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...
7
votes
2answers
132 views

Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the plane so that each arc is crossed at most once by another arc. A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times. Planar graphs are ...
0
votes
0answers
28 views

number of bipolar orientations to acyclic ones

Given a $k$-regular graph $G$, is there an upper bound on the number of bipolar orientations that $G$ has? I am trying to show that the number of bipolar orientations is much much lower than the ...
2
votes
1answer
57 views

Graph construction to double coloring & Hadwiger number

For any graph $G$ let $\eta(G)$ be the Hadwiger number of $G$. Is there for every graph $G$ a graph $2G$ such that -- $\chi(2G) = 2\chi(G)$, and -- $\eta(2G) = 2\eta(G)$? For each one of the above ...
3
votes
0answers
79 views

Independence Number of K4-free planar graphs

The independence ratio is defined as the size of the maximum independent set divided by $n$. By the 4-color theorem, every planar graph with $n$ vertices has independence ratio at least $1/4$. ...
10
votes
2answers
179 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 ...
5
votes
1answer
325 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 ...
13
votes
3answers
663 views

Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant of the integer lattice $\mathbb{Z}^2$, and from each site $s \in S$, one connects $s$ to every lattice point to which it has a ...
3
votes
1answer
65 views

How to prove a certain connection between the list-chromatic number of a bipartite graph and a cardinal?

Let $G=(L,R,E)$ be a complete bipartite graph, such that $|L|=\aleph_0$ and $|R|=\kappa$. I'd like to show that if $\kappa<2^{\aleph_0}$ then the list-chromatic number of $G$ can't be more than ...
2
votes
1answer
184 views

Coloring of subgraphs of G^n

Let $G=(L,R,E)$ be a finite bipartite graph, such that for each $v\in L\cup R: deg(v)>0$. Define $E^{(n)}=\{(\overline{l},\overline{r}) | \overline{l}=(l_1,...,l_n)\in L^n , ...
3
votes
0answers
77 views

Size of b-matching constructed from b maximal matchings

The Question: Let $G$ be a (simple) graph and let $b\in\mathbb{N}$. Suppose that we have $b$ disjoint edge-subsets $M_1,\ldots , M_b$ satisfying the following condition: The set $M_1$ is a maximal ...
6
votes
1answer
246 views

What is the status of this strong form of Hedetniemi's conjecture?

In this question, a graph is a finite, undirected graph without loops or multiple edges, and a colouring of a graph is a proper vertex colouring. The product $G \times H$ of graphs $G$ and $H$ is the ...
10
votes
1answer
319 views

Coloring $K_n$ via edge-weight sums

This is a question inspired by and tangential to "A Question on 1, 2 ,3 Conjecture"—and certainly much easier! Suppose one assigns a random edge weight among $\{1,2,3,\ldots,k\}$ to each edge ...
16
votes
1answer
652 views

A Question on 1, 2 ,3 Conjecture

1, 2, 3 conjecture is well-known: If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers ...
5
votes
1answer
293 views

coloring in lattice

This is a mathematical question raised from engineering and physics: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
2
votes
1answer
102 views

Achieving the largest possible minimum spacing between vertices of the same color in an integer lattice

Consider an infinite integer lattice, or an infinite hexagonal lattice with unit length edges. Provided a set of $k$ possible vertex colors, is there a known largest possible minimum spacing that can ...
3
votes
3answers
208 views

Crystal structure, lattice, Graph and coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
1
vote
1answer
228 views

When does graph Laplacian have eigenvalue -1?

Consider an undirected graph $G$ with (symmetric) adjacency matrix $A \in \{0,1\}^{n \times n}$ and degree sequence $d = (d_i)$ where $d_i = \sum_{j} A_{ij}$. Assume that every node has degree at ...
7
votes
0answers
148 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 ...
20
votes
3answers
1k views

Can one measure the infeasibility of four color proofs?

Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is ...
5
votes
1answer
267 views

Minimal graphs of prescribed girth and chromatic number

The well known result of Erdős, states that Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$ What I am wondering is When can we ...
5
votes
1answer
212 views

Probability that a random edge coloring of the complete graph is proper

This is a repost of this math.se question that I am posting here since it received no attention there. What is the probability that a random edge coloring of $K_n$ with $m \geq n$ colors ...
4
votes
2answers
126 views

Is There a Graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free

Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nesetril and Rodl proved that for a graph $G$, there exists a graph $F\to G$ with ...
1
vote
0answers
85 views

3-edge-coloring of 3-regular multigraphs

Given that a 3-regular multigraph is 3-edge-colorable, is there an expression for how many 3-edge-colorings exist? (For example, if a 2-regular multigraph is 2-edge-colorable, there are $2^k$ ...
5
votes
2answers
250 views

Fractional chromatic number, find reference to a particular alternate definition for

I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs. Let me review the most common definition and basic properties first. Let $ G $ be ...
1
vote
3answers
215 views

Strategic vertex labeling

We are given a graph $G=(V,E)$ with positive edge weights $w_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0(all vertices with ...
2
votes
1answer
279 views

bipartite graph coloring

Hi, I don't know much about graph theory so I would need to know if the following problem has a positive answer or a reference. There is a bipartite graph G with the two vertex sets V1, V2. Each ...
2
votes
0answers
121 views

Clique number of a $k$th power of a graph in terms of maximum degree?

Let $G$ be a graph of maximum degree $d$. Are there any known/easy bounds on the clique number of $G^k$, in terms of $d$ and $k$? I would like something at least epsilon better than the bound on the ...
1
vote
1answer
180 views

A yes no question concerning induced group

We are given a permutation group G acting on a finite set X. Finite set $\Omega$ contains all mappings $\omega$ from the set X to finite set K. We define mapping $\hat{g}$ on the set $\Omega$ in the ...
6
votes
0answers
245 views

Maximum fractional chromatic number of a 4-regular triangle-free graph (updated)

Let $G$ be a graph with maximum degree 4 and clique number 2. The fractional version of Reed's Conjecture tells us that $\chi_f(G) \leq 7/2$. But how high can $\chi_f(G)$ be? The Chvátal graph has ...
4
votes
1answer
458 views

Vector chromatic number and Lovasz theta

For $\alpha \ge 2$, an $\alpha$-vector coloring of a graph $X$ is an assignment of unit vectors to the vertices of $X$ such that vectors assigned to adjacent vertices have inner product less than or ...
6
votes
1answer
239 views

coincidence between minimal triangulation numbers and chromatic numbers

A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 ...
8
votes
3answers
421 views

Edge-coloring of the complete graph without any rainbow paths

For a given $2k-1$ edge coloring of the complete graph $K_{2k}$, say a Hamiltonian path $P$ is a rainbow path if every color appears exactly once in $P$. My question is "For each $2k (k \geq 2)$, is ...
2
votes
2answers
408 views

Coloring a graph by Maximum Independent Set extraction

recursively extracting a MIS from an undirected simple graph $G$ does produce a minimal coloring for $G$ ? I searched extensively the internet and found a paper [1] which answer partially to this ...