Genus bounds for embeddings with large minimum degree and representativity

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.

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size (where the constant depends on the connectivity), and even better bounds hold for planar graphs with minimum degree 4 and 5. In this paper, we investigate similar matching‐bounds for 1‐planar graphs, that is, graphs that can be drawn such that every edge has at most one crossing. We show that every 1‐planar graph with minimum degree 3 has a matching of size at least , and this is tight for some graphs. We provide similar bounds for 1‐planar graphs with minimum degree 4 and 5, while the case of minimum degree 6 and 7 remains open.

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

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).

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

We give an upper bound on the largest eigenvalue of the signless Laplacian matrix of a Hamiltonian graph. This bound is applied to obtain sufficient spectral conditions for the non-existence of Hamiltonian cycles. Under certain additional assumptions we provide a polynomial time decisive spectral criterion for the Hamiltonicity of a given graph with sufficiently large minimum vertex degree.

A note on the vertex-distinguishing proper coloring of graphs with large minimum degree

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 ${n\over k-1}$.

On the fixed edge of planar graphs with minimum degree five

Partitions of graphs and multigraphs under degree constraints

Purpose: The purpose of this paper is to prove the conjecture which states that the (1, 2 )-intersection index of a graph with () 3 is equal to zero.Moreover, practical applications of this result are given.Design/methodology/approach: We prove the conjecture by considering cases and indicating in each case two disjoint sets such that one of them is a (1, 1)-dominating set and the second one is a proper (1, 2)-dominating set.We also use a graph to model the problem of storage of supplies in Poland in case of a crisis.Findings: For every connected graph, in which every vertex has at least three neighbors, the (1, 2 )-intersection index is equal to zero.It ensures that in most cases there exists an optimal allocation of water and food supplies throughout a given region.Practical implications: The findings may be used by crisis management services when planning the allocation of food and water reserves.Originality/value: The obtained results and applications are new and original, they may be of value to mathematician working in the field of theoretical and applied graph theory, as well as to researchers working on crisis management and methods of efficient supply allocations.

Structure of edges in plane graphs with bounded dual edge weight