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, ...

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86 views

Properties of a smallest tournament with domination number $k$

For some tournament $T$, let $\gamma(T)$ denote the cardinality of a smallest dominating set of $T$. Denote by $f(k)$ the minimum number of vertices of a tournament $T$ having $\gamma(T) = k$. From ...
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209 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 ...
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53 views

Regular graphs with unimodal subdegrees that are not distance-regular

Distance regular graphs are known to exhibit the following property: starting from an arbitrary vertex $\alpha$, let $k_i$ denote the number of vertices at distance $i$ from $\alpha$ (in terms of ...
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80 views

Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...
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94 views

What is the projective dual of a planar graph?

Everybody learns the usual definition of the dual of a planar graph when edges are preserved and faces are mapped to vertices. Everybody learns the projective duality. What if we apply it to a ...
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61 views

Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...
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95 views

Proofs about Forbidden Minors

I am working on a problem which involves proving that a particular graph is a forbidden minor of the class of graphs that i am working on. Now i read kuratowskis theorem for planarity but i still ...
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117 views

Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure between two structures (let's say between two undirected simple graphs)? I mean, not "the measure with universal parameter that we ...
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57 views

Minimal set of 2-2 Pachner move null sequences on a (nonplanar) trivalent graph?

A "null sequence" is of course a sequence of Pachner moves (inside a closed area) that doesn't change the trivalent graph. E.g. doing the same Pachner move twice (leads to orthogonality of 6j symbols) ...
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92 views

Counting regular Hypergraphs

The problem of counting regular graphs on $n$ vertices is notoriously hard. It seems like counting regular hypergraphs on $n$ vertices should be much easier (I am placing no uniformity condition). ...
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108 views

Hamiltonian Matroids

Similar to graphs, a Matroid $M$ is said to be Hamiltonian if there is a base $B$ of $M$ and $e \in M-B$ such that $B + e$ is a cycle of $M$. Is there any literature on this? EDIT: Actually my ...
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90 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 ...
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153 views

The Turán problem for graphs with given chromatic number

The ordinary Turán problem for graphs asks, "Given a graph $H$, if $G$ is an $H$-free graph on $n$ vertices, what is the largest number of edges that $G$ can have?" As is well known, if $\chi(H) = r ...
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247 views

On the existence of Graph Monomorphism

A graph monomorphism is an injective graph homomorphism. Determining existence of Graph monomorphism between graph pairs is computationally hard. Assume we talk only about classes of undirected ...
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120 views

Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$. The bipartite independence number of ...
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99 views

Groups acting on non-locally-finite trees with independence and specified local actions

Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...
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292 views

Why the number of the vertices is even?

Let $G$ be a graph with $n$ vertices and $V$ the vertex set. Suppose that for each non-empty subset $W$ of $V$ there exists an element $\omega \in V$ ( maybe in $W$ or not) such that the degree of ...
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98 views

A surjective graph homomorphism between 4-valent graphs

Let $K$ and $L$ be $4$-valent planar graphs with Euler circuits (where the choice of the next edge in the circuit consists of the edge which does not share a face with the edge before), with $|V(K)| = ...
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97 views

Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...
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79 views

Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$). Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$. ...
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127 views

Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree? I would like to use semidefinite programming for ...
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473 views

Is there a name for this graph?

I'm trying to find out whether the following graph has a name: Let $W$ be an $n$-dimensional vector space over $GF(q)$. The vertices of the graph are all the subspaces of $W$. Two subspaces $W_1$ and ...
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302 views

Incremental minimum spanning tree

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq ...
2
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90 views

Shortest loop containing 0 in continuum percolation

I am interersted in continuum percolation with intensity $\lambda>0$. Formally, let $X$ be a Poisson point process in $\mathbb{R}^d$ with intensity $\lambda$ and $G$ the graph obtained by ...
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156 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 ...
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316 views

Partitioning the vertex set of a graph with a large independent set

Let $G$ be a graph on an even number of vertices, say $2M$. Assume that the largest independent set in $G$ has at most $M$ elements. Is it true that there exists a set of $2m$ vertices (for some ...
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73 views

Minimum relay nodes in graph

I have the following problem. I would like to know if this reduces to some standard problem in Graph theory. Any suggestions are much appreciated. I have a multi-cast network with 1 source (denoted ...
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152 views

Geodesics in polyhedral graphs

Let $e = \lbrace u,v\rbrace$, $e' = \lbrace v,u'\rbrace$ be edges of an undirected graph $G$ and $ee'$ be the path from $u$ through $v$ to $u'$. The following defintions make sense for every graph ...
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62 views

Presentation of tree decompositions (and related concepts) in terms of continuous maps?

A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure: Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$; The union ...
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76 views

Maximizing the minimum outdegree of digraph without $m$ cycle

Let $G$ be a simple digraph on $n$ vertices without a directed cycle of length $m$ (it may have directed cycles of length less than $m$. The cycles need not be simple). How large the minimum ...
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162 views

Structure of almost all bipartite graphs

I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true ...
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169 views

A primal-dual (double) circle packing (coin graph) question

I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem). Q1- But I am not sure whether ...
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168 views

What is the genus of a Johnson graph?

The Johnson graph $J(n, k)$ has a known genus when $k=1$, in which case it is the complete graph $K_n$. What can be said about the genus of $J(n, 2)$, or more generally $J(n, k)\ ?$
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118 views

Perfect P6-free graphs with further properties

Let $G$ be a graph without any hole or antihole of odd length at least 5 (i.e. $G$ is a Berge graph and so by the Strong Perfect Graph Theorem, $G$ is perfect). Assume further that $G$ has no ...
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155 views

Polyhedral embeddings of large face-width where all faces have the same length

Where can I find examples of polyhedral embeddings of simple graph with large face-width, such that all the faces have the same length? By polyhedral embedding I mean an embedding of the graph on a ...
2
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147 views

Additional Constraint Baum Welch for HMMs

I'm trying to derive a special form of the Baum Welch algorithm where there is an additional constraint that the sum of emission probabilities over all states sums to one for each output symbol. ...
2
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104 views

Graph theory: subset-simulation. Has already been studied under different names?

Hello, this is my first question, I hope it will be clear and correct enough. I am looking for a way to compare graphs in order to create a partial order among them (based on some kind of subgraph ...
2
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403 views

Matching in regular bipartite graphs

Let $G=(X,Y,E)$ be an $r$-regular bipartite graph where $|X|=|Y|$ . Let $\phi$ an automorphism of $G$ with $\phi(X) = Y$ and $\phi \circ \phi = id$, and let $\psi$ be the mapping induced by $\phi$ on ...
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333 views

Matrix Operations Preserving Hurwitz Stability

I begin with terminology I use in the question. A real square matrix $A$ is negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$; $\ast$-negative-stable ...
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94 views

Graphs, non-Hausdorfness and Wallman compactifications of non-regular spaces

Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) ...
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72 views

Subgraphs of bounded tree-width and preserving edges of original graph

Given a graph $G$, I would like to determine a method for randomly generating subgraphs $G'$ with the following properties: Each edge of $G$ has at least some probability $p$ of going into $G'$ The ...
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81 views

A non-distinct system of representative edges.

I have the following problem: Let $ \mathcal{G} = (G_{i})\_{i} $ be a collection of graphs. I would like to find a "system of representative edges" $ f : \mathcal{G} \rightarrow \bigcup_{i} E(G_{i}) ...
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252 views

Slightly improving bounds on two-color Ramsey numbers by globally pruning edges and counting connected vertices in instances of two-colored complete graphs

The two-color Ramsey number, $R(m, n)$, is the minimum number of vertices, $||V||$, in a complete graph necessary for there to exist a clique of order $m$ or an independent set of order $n$. In terms ...
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184 views

Maximum number of 4-colorings of planar graphs (precise version)

This is a redo of my earlier question. I'm trying again with more precision. I ignored one of the dicta in the "how to ask" page. For a fixed $n$, what is known (references preferred) about the ...
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233 views

Gromov-Hausdorff convergence for locally finite spaces

Update: I've edited the question, since maybe it was a bit confusing and it's better to start with a more basic question. I'm looking for properties of Gromov-Hausdorff convergence in the particular ...
2
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125 views

$f$-vector of graph connectivity

For a connected graph $G$, let $N_i$ be the number of connected subgraphs of size $i$. The vector $\langle N_0, N_1, \dots \rangle$ is also known as the $f$-vector for the graph. As a superset of a ...
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241 views

Complexity of bipartite graphs and their matchings.

My question concerns a hypothetical family of bipartite graphs, $G_i$. Each graph $G_i$ has $2^i$ red nodes and $2^i$ blue nodes - so nodes get labelled by their color and a binary string of ...
2
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378 views

Hamiltonian paths in subgraphs of rectangular lattice graphs

Is following decision problem NP-hard / NP-complete: Having vertex-induced subgraph of rectangular lattice graph determine if any Hamiltonian path exists Having vertex-induced subgraph of ...
2
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0answers
181 views

Counting the independent sets of a comparability graph with maximum degree 3. Is it #P−complete ?

This question was first asked here: cstheory.stackexchange... Here we want to count the number of independent sets of a comparability graph. For Δ(G)=2 , i.e. the maximum degree of a vertex in G is ...
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191 views

Complexity of finding disjoint 2-factors with equal cardinality in cubic graphs?

Finding a connected 2-factor that contains every node in cubic graphs is $NP$-complete since it is equivalent to the Hamiltonian cycle problem. I'm interested in the complexity of finding vertex ...