Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.

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Help on the following extremal problem?

An induced cycle is a cycle that is an induced subgraph of G; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes. Can anyone please tell me what ...
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2answers
949 views

Largest graphs of girth at least 6

Let $e_6(n)$ be the greatest number of edges in a simple graph with $n$ vertices and girth at least 6. Let $G_6(n)$ be the set of simple graphs of order $n$ with girth at least 6 and $e_6(n)$ edges. ...
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273 views

graphs with maximal number of paths of given length

Hi, For a given number of edges, the non directed graph which maximises the number of paths of length 2 is the quasi-star or the quasi-complete graph. Does anyone know : 1- what is the non directed ...
17
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2answers
857 views

Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...
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1answer
300 views

Degree conditions for k-factor

I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, ...
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215 views

Almost all graphs have a subgraph from a large class of graphs with constant order

I will pose the question in relation to trees but the more general question that can be deduced from the title of this post is also very interesting. I suspect the question might have a very trivial ...
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1answer
470 views

Induced Paths of Order 4

In a graph $G=(V,E)$ of order $n$, what fraction of the $\binom{n}{4}$ $4$-subsets of $V$ can induce the path of order four? I looked at this question 30 years ago and was never able to come up with ...
9
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453 views

What is the correct statement of this Erdös-Gallai-Tuza problem generalizing Turan's triangle theorem?

In Zsolt Tuza's Unsolved combinatorial problems I, Problem 46 is the following conjecture: Let $G$ be a graph on $n$ vertices. Let $\alpha_1$ be the maximum number of edges of $G$ such that every ...
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3answers
529 views

Non-isomorphic graphs with the same numbers of closed walks

Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that: $1)$ $G_i‎\ncong H_i$ for $i=1, 2, \cdots, n$ $2)$ $...
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428 views

Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
2
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1answer
652 views

If many triangles share edges, then some edge is shared by many triangles

Let $G=(V,E)$ be a graph. Let $t$ denote the number of triangles in the graph, and $x$ denote the number of pairs of distinct triangles that share an edge. (For example in $K_4$ we have $t=4$ and $x=6$...
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718 views

Number of Geodesic Paths Passing Through a Vertex in an Expander Graph

Let $\{G_{n}\}$ be a sequence of $k$-regular expander graphs. For each $n$ assume that each pair of nodes in the graph is transmitting a unit load of traffic and the traffic goes through the minimum ...
8
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1answer
398 views

What is the state of the art for the Turán number of $K_{4,4}$?

In Chung and Graham's "Erdős on Graphs: His legacy of unsolved problems," they discuss several open problems concerning Turán numbers for bipartite graphs. There is a construction which gives graphs ...
2
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0answers
127 views

Smallest size for an incomplete tournament with property $S_k$

By a well-known probabilistic argument due to Erdos, if $k>1$ is an integer then for all large enough $n$, there an asymmetric relation $R$ on $X=\lbrace 1,2, \ldots ,n \rbrace$ (i.e. $R \subseteq ...
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1answer
636 views

Connected components of large induced subgraphs of hypercubes

Let $H$ be the $n$-dimensional hypercube, i.e. $\{0,1\}^n$ with edges between two vertices if and only if they differ in exactly one co-ordinate. We say that an edge is in direction $i$ if its ...
6
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1answer
612 views

Characterization of infinite paths in graphs

First an introduction. A directed graph we all know what is, and a graph is serial whenever every vertex has a successor. I do not consider the empty graph. A pair $(\mathcal{G},s)$ is called a ...
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371 views

An extremal problem for graphs having every edge contained in a 4-clique

This is a follow-up to Graphs with many triangles but few complete graphs on 4 vertices I'm looking for an upper bound for the difference between the number of edges and the number of 4-cliques in a ...
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451 views

Graphs with many triangles but few complete graphs on 4 vertices

Let $G$ be a graph on $n$ vertices with $an^2$ edges containing at most $an^2/2$ copies of $K_4$. If there are cubically many triangles, say $cn^3$, then there is at least one edge that is not ...
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1k views

A graph with few edges everywhere

Given a graph $G(V,E)$ whose edges are colored in two colors: red and blue. Suppose the following two conditions hold: for any $S\subseteq V$, there are at most $O(|S|)$ red edges in $G[S]$ for any ...
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1answer
386 views

existence of triangle-free graphs for sparse regular graphs of degree at most n/2

It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to the half of the number of ...
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1answer
282 views

Graphs far from being a collection of bicliques

A biclique is a complete bipartite graph. A graph is a "biclique collection" if it can be decomposed into the disjoint union of bicliques. Denote the set of such graphs by $\mathcal{BCC}$. Given a ...
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Online Library of Unlabeled Connected Graphs on n Vertices

Does anyone know of the link to an online library of of unlabeled, connected graphs on n vertices? I remember looking at such an archive a few years ago while at a Macaulay 2 workshop, but I've been ...
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372 views

Cliques of hyperedges

Suppose we have a graph, with multiple edges allowed. An edge-clique is a set $C$ of edges so that every two edges in $C$ share at least one endpoint. Note that any edge-clique falls into one of two ...
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473 views

Bounds on the independence number of a graph

If $G$ is a graph with $n$ vertices and $\frac{nk}{2}$ edges, $k\ge -1,$ then $a(G)\ge \frac{n}{k+1}$. Why? (Here $a(G)$ is the independence number).
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Is the feedback vertex number bounded by the maximum number of leaves in a spanning tree?

I have a graph-theoretical conjecture which I think would have been studied before, but for which I cannot find anything in the literature. Let G be a finite, simple, connected graph. Let the ...
2
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1answer
392 views

Suppose the independent number of a graph is bounded. How small the clique number can be?

Suppose the independent number of a graph is bounded. How small the clique number can be? linear? It seems to be a natural problem to ask. but I could not find any reference. Thanks.
2
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2answers
512 views

Minimal Non-planar Extensions of a Graph

Given a planar graph $G=(V,E)$ with vertices $V$ and edges $E$, call $\bar G = (V,\bar E)$ a non-planar extension of $G$ if $\bar G$ is non-planar and $E \subset \bar E$. I'm interested in minimal ...
5
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3answers
911 views

Erdős–Stone theorem type edge density estimates for bipartite graphs?

The Erdős–Stone theorem theory says that the densest graph not containing a graph H (which has chromatic number r) has number of edges equal to $(r-2)/(r-1) {n \choose 2}$ asymptotically. However, ...