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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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        0answers
        18 views

        Can you create a directed graph on five vertices where each vertex touches every other vertex in one or two moves [on hold]

        Can you create a directed graph on five vertices where each vertex touches every other vertex in one or two moves.
        2
        votes
        1answer
        69 views

        How does the complexity of calculating the Permanent imply the NP completeness of directed 3-cycle cover?

        In their paper Two Approximation Algorithms for 3-Cycle Covers of Markus Bl?ser and Bodo Manthey it is stated that: "...deciding whether an unweighted directed graph has a 3-cycle cover is already NP-...
        1
        vote
        0answers
        38 views

        Coloration of an interval graph with constraints [on hold]

        Given an interval graph that represents a set of tasks, in a given period of time, to be assigned to a set of employees, the objective is to find a minimum coloring of this graph such that the total ...
        0
        votes
        0answers
        34 views

        Algorithmic complexity of deciding the existence of regular $\mathrm{f}$-factors in graphs

        Finding regular $\mathrm{f}$-factors in undirected simple graphs can be reduced to finding a perfect matching by utilizing the gadgets of Tutte or of Lovász and Plummer; there are several algorithms ...
        4
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        1answer
        141 views

        Probability of a vertex being a “degree-celebrity” in a random graph

        If $G(n,p)$ is a random graph of the Erd?s-Rényi model, what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$ Please feel free to relate answers to other ...
        -2
        votes
        0answers
        45 views

        Polya Enumeration to study proper graph colorings

        Can Polya enumeration technique or its modification be used in evaluating the chromatic polynomial of a simple graph? Specifically, we have to ensure that two distinct points have distinct colors if ...
        2
        votes
        0answers
        76 views

        Efficiently computable graph conductance measures

        Treating electrical networks from a graph theory point of view, do there exist measures that characterize the overall electrical conductance of the graph whilst being efficiently computable? There ...
        8
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        0answers
        160 views

        Thurston on the Robertson-Seymour theorem

        Danny Calegari recounts here that Bill Thurston "gave one talk explaining his idea of a new proof of (some version of) the Robertson-Seymour theorem" at MSRI in 1996-1997. I could not find it in the ...
        4
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        0answers
        199 views

        Reference for results about planar graphs

        A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
        2
        votes
        1answer
        122 views

        Is this graph problem NP-Hard?

        I had asked this question in math.se without any success Let $A$ be the symmetric $n\times n$ adjacency matrix for a graph where $A_{ij}$ is the positive edge value between node $i$ and $j$ (thus ...
        5
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        1answer
        160 views

        Random walk on the hypercube with deleted edges

        Let $G$ be the $n$-dimensional boolean hypercube, i.e. the graph on $\{0,1\}^n$ where two vertices are adjacent iff they differ on exactly one coordinate. Consider a graph $G'$ obtained by deleting a ...
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        3answers
        713 views

        When can a graph be oriented to form a Hasse diagram of a finite poset?

        For any finite poset $P=(X,\leq)$ there is an undirected graph $G$ underlying the Hasse diagram of $P$ such that $V(G)=X$ and $E(G)=\{\{u,v\}:u\lessdot v\}$. With that said, is it possible to ...
        2
        votes
        1answer
        92 views

        Electrode assignment problem in resistive networks

        Main question In the context of resistor networks, and particularly purely from a graph theory point of view, is there a consistent way of assigning the two electrode nodes in order to compare the ...
        8
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        0answers
        235 views
        +50

        Connected subgraphs and their sums

        Let $G=(V,E)$ be an undirected graph with $|V|\geq 4$ such that for any distinct vertices $a_1,a_2,b_1,b_2$, there is a path from $a_1$ to $a_2$ and a (vertex-)disjoint path from $b_1$ to $b_2$. ...
        2
        votes
        1answer
        119 views

        Edge coloring graphs is in P?

        It is known that there exist polynomial time algorithm to approximate the Lovasz number or the supremum of Shannon capacity of graphs. By Vizing's theorem, the graph $G$ has only two chromatic ...
        1
        vote
        1answer
        58 views

        Perfect graphs condition could be weakened?

        The perfect graphs are generally defined as those graphs whose every induced subgraph has its chromatic number equal to its clique number. Now,are there some examples where the clique number of graph ...
        1
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        0answers
        81 views

        Chromatic number of certain graphs with high maximum degree

        Let $G$ be the graph of even order $n$ and size $\ge\frac{n^2}{4}$ which is a Cayley graph on a nilpotent group but not complete. Can the chromatic number of this graph be determined in polynomial ...
        20
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        0answers
        464 views

        Is this representation of Go (game) irreducible?

        This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph. Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
        2
        votes
        1answer
        66 views

        Define a homomorphism of a set of graphs to its power set

        Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is, $G_1\cup G_2$ $=\langle V(G_1)\cup V(G_2), (E(G_1)\...
        1
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        0answers
        51 views

        Combinatorial equation system with exponentially many equations in quadratic many variables

        A certain question on graph theory (about the existance of graphs with a certain coloring inherited by perfect matchings) can be translated into the satisfiability problem of a certain set of ...
        2
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        0answers
        177 views

        Is there a known proof that $R(5,5)\leq 47$ in Ramsey theory?

        As an application to a model describing graphs with partial information, I found what might be an (as yet unverified) proof that $R(5,5)\leq 47$. According to the Dynamic Survey of Ramsey Numbers at ...
        3
        votes
        1answer
        47 views

        Connected hypergraphs

        We say that a hypergraph $H=(V,E)$ is connected if the following condition holds: for all $S\subseteq V$ with $\emptyset\neq S \neq V$ there is $e\in E$ that meets both $S$ and $V\setminus S$, i.e. ...
        6
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        0answers
        105 views

        Squared squares and partitions of $K_{nn}$

        This is inspired by a recent question. Define a square square sum (SSS) of order $n$ to be any partition $$n^2=\sum_1^tc_ii^2 \tag{*}$$ of $n^2$ into square summands. Call it perfect if all $c_i \leq ...
        1
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        0answers
        81 views

        Digraphs with same number of semiwalks

        This is a follow-up question to Characterisation of walk-equivalent digraphs. Question: Do there exists two directed graphs $G$ and $H$ consisting of the same number ($n$) of vertices, such that \...
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        0answers
        35 views

        Generalization of Menger's Theorem to Infinite Graphs

        Aharoni and Berger generalized Menger's Theorem to infinite graphs: For any digraph, and any subsets A and B, there is a family F of disjoint paths from A to B and a set separating B from A consisting ...
        1
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        0answers
        25 views

        Worst case performance of heuristic for the non-eulerian Windy Postman Problem

        The Windy Postman Problem seeks the cheapest tour in a complete undirected graph, that traverses each edge at least once; the cost of traversing an edge is positive and may depend on the direction, in ...
        1
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        0answers
        22 views

        Definition of k-partite hypergraph

        I would like to know the standard definition of k-partite hypergraph. There are two natural generalizations of k-partite graph to k-partite hypergraph: 1. For all edges e, any two vertices in e are ...
        15
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        7answers
        972 views

        Examples of proofs by making reduction to a finite set [closed]

        This is a very abstract question, I hope this is appropriate. Suppose $T$ is some claim over some infinite set $A$, for example, let $A$ be the set of all loopless planar graphs, and $T$ be the claim "...
        1
        vote
        1answer
        88 views

        Characterisation of walk-equivalent digraphs

        Setting Let $G=(V,E)$ be an undirected graph. A walk $\pi$ in $G$ of length $k$ is a sequence of $k+1$ vertices $v_1,\ldots,v_{k+1}$ such that for each $i\in[1,k]$, $\{v_i,v_{i+1}\}\in E$. Let $H=(W,F)...
        0
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        0answers
        38 views

        Graphs “weak” in context of cutting subgraphs

        Lately I've been looking into graphs (simple, undirected, finite) that are in some way weak when it comes to connectivity, that is: Let $G$ be a graph of order $n$. We'll say that $G$ is $k$-weak if ...
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        0answers
        31 views

        Treewidth related properties of a bipartite graph with bounded local crossing number and diameter

        If a bipartite degree at most $3$ graph on $O(n^2)$ vertices with diameter at most $O(\log n)$ has property that every edge intersects at most $O(\log n)$ edges on a planar drawing then does any of ...
        2
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        0answers
        220 views

        Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?

        In the following paper (extended version here), at the beginning of section 3, the authors give two axioms about $\mathcal{A}$. Axiom 1 is about $\mathcal{A}$ being an algebra. I do not see where this ...
        3
        votes
        1answer
        156 views

        Diameter of Cayley graphs of finite simple groups

        Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article). THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
        11
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        4answers
        452 views

        A specific collection of subgraphs in $K_{70, 70}$

        Does there exist a collection of subgraphs $\{\Gamma_i\}_{i = 1}^{24}$ of $K_{70, 70}$, that satisfy the following two properties: 1)$\Gamma_i \cong K_{i, i} \forall 1 \leq i \leq 24$; 2)Any ...
        4
        votes
        0answers
        77 views

        Dinitz Conjecture extension to rectangles

        The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in ...
        2
        votes
        1answer
        77 views

        List coloring of tripartite graph [closed]

        Let $G$ be a tripartite graph with partite sets $A,B,C$. The graphs $A\cup B$, $B\cup C$ and $C\cup A$ are each bipartite. Let the maximum degree of the graph be $\Delta$. Now, we know that the ...
        0
        votes
        0answers
        9 views

        Iterated Inverse structures: polynomial representation of integer partitioning of preimages in Sigma Matrices (reference request)

        I am studying iterated preimage structures of functions on a finite set. The main structure of interest to me, the Sigma Matrix, is derived from a matrix listing the element-wise preimage sets at ...
        0
        votes
        1answer
        71 views

        If the core of a graph is a forest, then it is Class 1

        It is a standard result, due to Fournier, that if the core of a graph (the induced graph by the vertices having their degrees equal to maximum degree of the graph) is a forest (acyclic), then the ...
        1
        vote
        1answer
        46 views

        Number of occurrences of subgraphs as a unique identifier

        Given $q \in \mathbb{N}$, let $B_q$ be a sequence of all (non isomorphic) connected graphs with at most $q$ vertices. Now for a given connected graph $G$, lets define signature of $G$ ($sig_q(G)$) as ...
        7
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        0answers
        161 views

        Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?

        Let $A$ be a $2n$-by-$2n$ matrix with entries in $\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$ has rank $\...
        1
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        0answers
        60 views

        Is there a well-posed definition of game on a graph? Or a well defined category of games on graphs?

        All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions: 1) Is there a well-posed mathematical definition of game on ...
        1
        vote
        0answers
        29 views

        Determining the minimum weight maximal oriented subgraph of a complete directed graph

        Let $G(V,A,W):\ |V| = n,\ A=V\times V\setminus \lbrace (v,\ v)\rbrace,\ W\in\mathbb{R}_+^{n\times n},\ W^T\ne W $ be a complete directed graph with asymmetric weights. Questions: What is ...
        0
        votes
        0answers
        58 views

        Infimums of Poset of Unlabelled Subtrees

        I will use $T$ to refer to the set of unlabelled, rooted trees, and use $(t,r)$ to denote a tree and its root. Let $(T, \preceq)$ be a poset where $(t_1,r_1) \preceq (t_2,r_2)$ means that $t_1$ is a ...
        2
        votes
        1answer
        77 views

        Two cospectral (normal) digraphs which are not orthogonal similar

        Preliminaries A complex matrix $A$ is normal when $A$ and $A^*$ commute. A real matrix $A$ is normal when $A$ and $A^t$ commute. Two complex matrices $A$ and $B$ are said to be unitary similar if ...
        4
        votes
        3answers
        230 views

        Is there a name for this “stack” of graphs?

        Let $G_1,\ldots,G_m$ be a sequence of graphs, all having the same number $n$ of vertices. For each pair $(G_i, G_{i+1})$ we add $n$ edges that connect the vertices of $G_i$ and $G_{i+1}$ bijectively. ...
        1
        vote
        1answer
        132 views

        Quotient graph of a tree

        We know that every graph is isomorphic to a subgraph of a complete graph. Similarly, can we say that every graph is isomorphic to a quotient graph of a tree?
        0
        votes
        1answer
        57 views

        Combining three matchings to form a maximal matching

        Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite. Now, is there a way to ...
        1
        vote
        1answer
        108 views

        Number formation and bridged graphs, connection or coincidence?

        Bridged graphs sequence $g(n) =$ "Number of simple connected bridged graphs on $n+2$ nodes". We have $g(n)=1, 3, 10, 52, 351, 3714,\dots$ from A052446. Number formation sequence We also have $f(n) ...
        6
        votes
        1answer
        67 views

        Minimizing the number of segments in drawings of planar graphs

        Every planar graph has at most $3n-6$ edges, where $n$ is the number of vertices. Moreover, every planar graph can be drawn with straight-line edges in the plane, without crossings. For example, for ...
        1
        vote
        0answers
        128 views

        Research-level blogs on complex networks:

        I'm an applied mathematician that has a research interest in complex networks for modelling biological systems and I wondered whether the MathOverflow community might know of research-level blogs that ...

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