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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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### Bounds on chromatic number when maximum degree is large

For a regular graph with $n$ vertices and maximum degree $\Delta$, it is easy to see that the chromatic number, $\chi\le\frac{n}{2}$ if $\frac{n}{2}\le\Delta\lt n-1$(since a regular graph on $n$ ...
115 views

### Chromatic number of the linear graph on $[\omega]^\omega$

Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$. Let $$E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\}.$$ It is clear that $G = ([\omega]^\omega, E)$ has no ...
39 views

### Clarifications regarding conformability in graph colorings

As an outgrowth of this question, I have another question, that is, why not the definition of conformability includes a $\Delta$ vertex coloring also, instead of only $\Delta+1$ coloring of vertices. ...
78 views

### A simple case of a strong version of the Berge-Fulkerson conjecture

UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ...
19 views

### Perfect matchings and edge cuts in cubic graphs - part 1

Let $G$ be a bridgeless cubic (simple) graph, and let $M$ be a perfect matching in $G$. $G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $K_{3,3}$ we ...
124 views

### Graceful graphs all of whose vertices are labelled with primes or squares

Do graceful graphs exist with more than any arbitrarily large number of vertices, all of which are labelled with a prime or non-negative square number. Recall that a graceful graph is a graph with m ...
117 views

### Has anyone implemented a circle graph recognition algorithm?

A double occurrence word is a circular string of length $2n$ over an alphabet of size $n$ with each letter occurring exactly twice, for example: ABACCDBD Given a ...
137 views

### Structure of color critical graph

Let $G$ be a $k$-color critical graph on $N$ vertices. It is a known fact that every vertex of $G$ has at least $k-1$ neighbors (there are more results available on minimum number of edges in color ...
95 views

### Upper bounds for the second largest eigenvalue in terms of degree?

I am looking for upper bounds on the second largest eigenvalue, $\lambda_2(G)$ of a given graph $G$, with respect to minimum/maximum degrees of the graph. I looked around for some existing bounds most ...
35 views

### Matching number versus Independence number in a graph

I am looking at partial functions f:V -> V on a finite graph (undirected no loops or multiple edges) G=(V,E) such that 1) The range f(V) is an independent set of vertices. 2) The "kernel" ...
60 views

### Upper bound on the length of chordless cycles in d-regular graphs

Given a $d$-regular graph with $n$ vertices is there a known (non-trivial) upper bound on the length of chordless cycles in it (presumably as a function of $d$ and $n$)? I wasn't able to find anything ...
97 views

Let $\langle V_1, E_1 \rangle, \langle V_2, E_2 \rangle$ and $\langle V_3, E_3\rangle$ be any three undirected simple graphs with $m$, $n$ and $p$ number of edges, respectively such that $E_2$ and $... 0answers 70 views ### How is the following graph operation defined in the given research paper? enter image description hereI am unable to understand the graph operation defined in the following paper https://www.sciencedirect.com/science/article/pii/S0972860017301706. Can someone kindly ... 0answers 122 views ### Generalized graph-minor theorem? Consider the following generalized graph-minor theorem: GM($κ,λ$): Given any collection$S$of$κ$simple undirected graphs each with less than$λ$vertices, there are distinct graphs$G,H$in$S$... 0answers 32 views ### Groups that can occur as graph automorphisms of a fixed size graph From theorem$4$and corollary$1\$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...

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