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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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### 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 ...
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 ...
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" ($b$), "...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 "...
88 views

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 $H=(W,F)... 0answers 38 views ### Graphs “weak” in context of cutting subgraphs Lately I've been looking into graphs (simple, undirected, finite) that are in some way weak when it comes to connectivity, that is: Let$G$be a graph of order$n$. We'll say that$G$is$k\$-weak if ...

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