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# Questions tagged [graph-theory]

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, topological-graph-theory, random-graphs, graph-colorings and several others.

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### Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points

Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...
48 views

### An upper bound on the minimum number of vertices in a girth 5 graph of chromatic number $k$

Is there a known upper bound on the minimum number of vertices in a graph with girth 5 and chromatic number $k$? Could you also give references for this?
37 views

### Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 14?

Consider a planar graph where every vertex is incident to at least 3 edges, and assign to each edge a weight equal to the sum of the degrees of its endpoints. If not, what is the smallest n so that ...
20 views

### Counterpart of dominating sets in graphs

A $t$-fold dominating set in a simple undirected graph $G$ with vertex set $V$ is a subset $D\subseteq V$ such that each vertex of $V\setminus D$ has at least $t$ neighbours in $D$. I am interested ...
33 views

### Induced subgraphs of the line graph of a dense linear hypergraph

Given a hypergraph $H=(V,E)$ we associate to it its line graph $L(H)$ given by $V(L(H)) =E$ and $$E(L(H)) = \big\{\{e_1,e_2\}: e_1\neq e_2 \in E \text{ and } e_1\cap e_2 \neq \emptyset \big\}.$$ We ...
42 views

### Rank and edges in a combinatorial graph?

Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
49 views

### Automorphism group of the cycle graph with $k$ diagonals

Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$. Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...
22 views

### Practical calculation of minimum weight vertex-disjoint cycle covers

How are minimum-weight vertex-disjoint cycle covers of large dense symmetric graphs actually calculated in actual implementations? I know that the problem can be reduced to general matching by ...
49 views

### Maping of subcubes of a $(d+k)$-hypercube onto subcubes of $d$-hypercube

Denote by $Q_n$ the n-dimensional hypercube. A vertex of $Q_n$ is represented by a vector of $n$ $\{0,1\}$-bits. An edge corresponding to two vertices whose vectors differ in one coordinate is ...
138 views

### What is the maximal number of triangles in a planar graph?

What is the maximal number of triangles in a planar graph with $n$ points? Here by triangle I mean any closed path of length 3 (so a triangle can also include in the interior points other than the ...
103 views

### Avoiding multiply covered vertices in graph edge coverings

Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an edge cover of $G$ if $\bigcup C = V$. For any edge cover $C$ ...
93 views

### Chromatic polynomial of a bipartite graph replaced by a new graph

Consider a semi-regular bipartite graph $C$ consisting of two parts $A$ (having each vertex of degree $\Delta$) and $B$ (having each vertex of degree $2$). Let its chromatic polynomial be $C(x)$. Now,...
55 views

### Proving Vizing's and Brooks' theorem using the polynomial approach

It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
23 views

### Product of edge monomials and chromatic number

As a continuation of this question and as stated in the comments, I wish to ask if the minimum value of $rad$ for all monomials equals the chromatic number when the number of variables in a monomial ...
113 views

### Graphs with Hermitian Unitary Edge Weights

Very recently, Hao Huang proved the Sensitivity Conjecture, which had been open for 30 years or so. Huang's proof is surprisingly short and easy. Here is Huang's preprint, a discussion on Scott ...

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