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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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      11
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      4answers
      358 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 ...
      6
      votes
      0answers
      84 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 ...
      0
      votes
      1answer
      88 views

      Any results concerning the numbers of vertices and edges to form fixed number of cliques in $K_n$?

      Given a complete graph $K_n$, and if we know there are $t$ $K_s$ ($s\ge 2$) in it, what can we say about the possible number $a$ of vertices and the number $b$ edges to form these $t$ cliques? We can ...
      0
      votes
      0answers
      47 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 \...
      3
      votes
      0answers
      31 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 ...
      13
      votes
      7answers
      835 views

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

      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
      0answers
      24 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
      vote
      1answer
      83 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
      votes
      0answers
      16 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 ...
      1
      vote
      1answer
      169 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 ...
      -7
      votes
      2answers
      302 views

      Do degrees determine the chromatic number?

      Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}_G(v) = \deg_H(\varphi(v)).$...
      9
      votes
      0answers
      228 views
      +50

      Correspondence between matrix multiplication and a graph operation of Lovasz

      In his book "Large networks and graph limits" (available online here: http://web.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf), Lovasz describes a multiplication operation (he calls it ...
      2
      votes
      1answer
      59 views

      List coloring of tripartite graph

      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 ...
      1
      vote
      2answers
      114 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 ...
      0
      votes
      0answers
      17 views

      What's the best method for sorting many multiple-characteristic items into multiple-characteristic bins? [on hold]

      ? There are 100 items to be sorted into just over 100 bins, where no bin has more than one item and not every bin is necessarily filled, but every item must be placed. ? There are certain items that ...

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