A survey of degree-boundedness

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

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 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 ${n\over k-1}$.

We prove the following extension of an old result of Andrásfai, Erdős and Sós. For every fixed graph $H$ with chromatic number $r+1 \geq 3$, and for every fixed $\epsilon&gt;0$, there are $n_0=n_0(H,\epsilon)$ and $\rho=\rho(H) &gt;0$, such that the following holds. Let $G$ be an $H$-free graph on $n&gt;n_0$ vertices with minimum degree at least $\left(1-{1\over r-1/3}+\epsilon\right)n$. Then one can delete at most $n^{2-\rho}$ edges to make $G$ $r$-colorable.

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.

We show that for every , there exists such that for every even , and every graph with vertices and minimum degree at least , each colouring of the edges of with three colours results in a monochromatic cycle of length .

In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. Ford≥ 2, letdbe the set of connected graphs of minimum degree at leastd. LetF1andF2be connected graphs and letbe a set of connected graphs. Then {F1,F2} is said to be a forbidden pair forif every {F1,F2}-free graph inof sufficiently large order has a 2-factor. Faudree, Faudree and Ryjáček have characterized all the forbidden pairs for the set of 2-connected graphs. We first characterize the forbidden pairs for2, which is a larger set than the set of 2-connected graphs, and observe a sharp difference between the characterized pairs and those obtained by Faudree, Faudree and Ryjáček. We then consider the forbidden pairs for connected graphs of large minimum degree. We prove that if {F1,F2} is a forbidden pair ford, then eitherF1orF2is a star of order at mostd+ 2. Ota and Tokuda have proved that every$K_{1, \lfloor\frac{d+2}{2}\rfloor}$-free graph of minimum degree at leastdhas a 2-factor. These results imply that fork≥d+ 2, no connected graphsFexcept for stars of order at mostd+ 2 make {K1,k,F} a forbidden pair ford, while for$k\le \bigl\lfloor\frac{d+2}{2} \bigr\rfloor$every connected graphFmakes {K1,k,F} a forbidden pair ford. We consider the remaining range of$\bigl\lfloor\frac{d+2}{2} \bigr\rfloor &lt; k &lt; d+2$, and prove that only a finite number of connected graphsFmake {K1,k,F} a forbidden pair ford.