# Tagged Questions

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, ...

72 views

### Existence a class of graphs with special property

In the following, suppose all graphs are simple and finite. For a given graph $G$, we denote its complement by $\overline{G}$. Let $*$ be a binary operation among graphs, such as Cartesian product, ...
140 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 ...
59 views

### Is transitive reduction for a direct acyclic graph really unique? [closed]

According to Wikipedia, "If a given graph is a finite directed acyclic graph, its transitive reduction is unique" Here is what I think might be a counter-example: Imagine a diamond-shaped DAG where ...
79 views

### Isomorphism with fixed number of Permutations [closed]

Suppose, I have a fixed number of permutations for each sub-graph to determine isomorphism of whole graph. Is it possible to determine efficiently ? For example, $G, H$ are isomorphic graphs. For ...
58 views

### A plane graph problem [closed]

Let G be a planar graph, with edges colored red and blue. Show that there is a vertex v such that going round the vertex in a clockwise direction we encountered no more than two change of colors. Has ...
76 views

66 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 ...
207 views

### Graphs with prescribed numbers of k-cliques

Let $(a_1,a_2,\dots, a_n)$ be a sequence of non-negative integers. Q. When does there exists a simple graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_1$ vertices, $a_2$ ...
77 views

### Triange-free graph and its complement has Lovász number > 3

I found an example by the method in the paper Explicit Ramsey graphs and orthonormal labelings by Noga Alon 1994. The graph is around $10^6$ vertices, anyone knows smaller graph which is Triangle-free ...
128 views

### Hadwiger number and minimal degree

Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximal $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree, do we have $\delta(G)\leq\eta(G)$?
24 views

### History of the Vertex Disjoint Cycle Cover with Minimal Edgeweight Sum

Questions: who first posed the problem of determining a collection of (directed) cycles, whose edgeweight sum is minimal and, for which each vertex belongs to exactly one of the cycles? who came up ...
54 views

### Hadwiger number of total graph

Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its total graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. Is there an ...
59 views

### Asymptotic formula for average geodesic length on graph?

If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length? That is suppose I know two nodes $n_1$ and $n_2$, the number of edges in my graph and at those ...
95 views

### Total chromatic number and total clique number

Let $G=(V,E)$ be a finite, simple, unconnected graph. We define the total graph $T(G)$ of $G$ as follows: $V(T(G)) = (V\times\{0\}) \cup (E\times\{1\})$, $E(T(G)) = E_v \cup E_e \cup E_{v+e}$, where ...
41 views

### Generating tournaments inductively

This is a somewhat vague question, but I'm interested in ways to create a strong tournament from one or more smaller tournaments. Obviously, the disjoint union of two tournaments is a new tournament, ...
106 views

69 views

59 views

### bounded degree graph colouring.

I was wondering if anyone could provide references on the following: Is determining the chromatic number of a bounded degree graph APX-complete? 2.I've seen the result that states it is NP-hard ...
73 views

### Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...