Matchings in 1‐planar graphs with large minimum degree

It is well known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalize the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.

Genus bounds for embeddings with large minimum degree and representativity

Graphs with large minimum degree containing no copy of a clique on $r$ vertices ($K_r$) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that $K_r$-free graphs $G$ with degree larger than $((3r-7)/(3r-4))|V(G)|$ must be $(r-1)$-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order $(1/(r-1))|V(G)|$. The following paper provides improved bounds on the minimum degree which would imply the same conclusion. This problem was first considered by Brandt, and we provide improvements over these initial results for $r > 5$.

Wiener index and Harary index on Hamilton-connected graphs with large minimum degree

A note on 1-planar graphs with minimum degree 7

For two graphs F, H and a positive integer n, the function fF,H(n) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where F is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer r and every Kr-1-free graph F, there exists some εF>00$$\\end{document}]]> such that fF,Kr(n)=O(n1/2-εF). This result is tight in two ways. Firstly, it is no longer true if F contains Kr-1 as a subgraph. Secondly, we show that for all r≥4 and ε>00$$\\end{document}]]>, there exists a Kr-1-free graph F for which fF,Kr(n)=Ω(n1/2-ε). Along the way of proving this, we show in particular that for every graph F with minimum degree t, we have fF,K4(n)=Ω(n1/2-6/t). This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants 0<c<C such that for each r≥4, if F is a bipartite graph with sufficiently large minimum degree, then Ω(nclogr)≤fF,Kr(n)≤O(nClogr). This shows that for graphs F with large minimum degree, the behaviour of fF,Kr(n) is drastically different from that of the corresponding off-diagonal Ramsey number fK2,Kr(n).

Structure of edges in plane graphs with bounded dual edge weight

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if for a function that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015

A survey of degree-boundedness

A note on Huang–Zhao theorem on intersecting families with large minimum degree

A lower bound for the smallest uniquely hamiltonian planar graph with minimum degree three

On the fixed edge of planar graphs with minimum degree five

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. (ICALP '12) introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree k the equilibrium for this model is a maximal subgraph whose minimum degree is at least k. However the dropping out of individuals with degrees less than k might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. considered the Anchored k-Core problem: Given a graph G and integers b, k and p do there exist sets of vertices B, H such that B is a subset of H, size of B is at most b and size of H is at least p, and every vertex v which is in H but not in B has degree at least k in the induced subgraph G[H]. They showed that the problem is NP-hard for all k greater equal 2, and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the Anchored k-Core problem is W[1]-hard parameterized by p, even for k=3. This improves the result of Bhawalkar et al. (who show W[2]-hardness parameterized by b) as our parameter is always bigger since p is greater equal than b. Then we answer a question of Bhawalkar et al. by showing that the Anchored k-Core problem remains NP-hard on planar graphs for all k greater equal 3, even if the maximum degree of the graph is k+2. Finally we show that the problem is FPT on planar graphs parameterized by b for all k greater equal 7.

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.