On the local density problem for graphs of given odd-girth

formula omitted]-restricted edge-connectivity in triangle-free graphs

Let G = ( V , E ) be a finite undirected graph with vertex set V ( G ) of order | V ( G )| = n and edge set E ( G ) of size | E ( G )| = m . Let Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ be the degree sequence of the graph G . A clique in a graph G is a complete subgraph of G . The clique number of a graph G , denoted by ω ( G ) , is the order of a maximum clique of G . In 1907 Mantel proved that a triangle-free graph with n vertices can contain at most ⌊ n 2 /4⌋ edges. In 1941 Tur'an generalized Mantel’s result to graphs not containing cliques of size r by proving that graphs of order n that contain no induced K r have at most (1 − 1/ r − 1) n 2 /2 edges. In this paper, we give new bounds for the maximum number of edges in a K r -free graph G of order n , minimum degree δ , and maximum degree Δ . We show that, for the families of graphs having the above properties, our bounds are slightly better than the more general bounds of Tur'an.

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

Triangle-free graphs whose independence number equals the degree

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Connectivity keeping paths in k-connected bipartite graphs

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\\operatorname{mc}(G) \\geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.\n¶ In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$ -approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Möbius ladder. A major tool is the observation that every triangle-free graph with δ[ges ]n/3 has a unique maximal triangle-free supergraph.

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index $$R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} $$ , where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for −1 ≤ α < 0, respectively.

The clique removal lemma says that for every and , there exists some so that every ‐vertex graph with fewer than copies of can be made ‐free by removing at most edges. The dependence of on in this result is notoriously difficult to determine: it is known that must be at least super‐polynomial in , and that it is at most of tower type in . We prove that if one imposes an appropriate minimum degree condition on , then one can actually take to be a linear function of in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

Network structure and minimum degree

Shortest Fuzzy Hamiltonian Cycle on Transportation Network Using Minimum Vertex Degree and Time-dependent Dijkstra’s Algorithm

We propose the following extension of Dirac's theorem: if is a graph with vertices and minimum degree , then in every orientation of there is a Hamilton cycle with at least edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree guarantees a Hamilton cycle with at least edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability is above the Hamiltonicity threshold, then, with high probability, in every orientation of there is a Hamilton cycle with edges oriented in the same direction.

We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V|-3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.