Formula omitted]-factors in graphs

An improved upper bound on the double Roman domination number of graphs with minimum degree at least two

Consecutive magic graphs

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 ${n\over k-1}$.

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.

Irregularity strength of dense graphs

3-component domination numbers in graphs

For a positive integer k, a graph G is k‐ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, then G is k‐ordered hamiltonian. Minimum degree conditions are also given for k‐ordered hamiltonicity. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 199–210, 2003

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$n?3 and W a subset of V(G) with $$|W|\ge 3k$$|W|?3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)?2n/3 for each $$x\in W$$x?W, then G contains k vertex-disjoint cycles such that each of them contains at least three vertices of W. In this paper, we obtain an analogue result of Wang's Theorem in bipartite graph with the partial degree condition. Let $$G=(V_1,V_2;E)$$G=(V1,V2?E) be a bipartite graph with $$|V_1|=|V_2|=n$$|V1|=|V2|=n, and let W be a subset of $$V_1$$V1 with $$|W|\ge 2k$$|W|?2k, where k is a positive integer. We show that if $$d(x)+d(y)\ge n+k$$d(x)+d(y)?n+k for every pair of nonadjacent vertices $$x\in W, y\in V_2$$x?W,y?V2, then G contains k vertex-disjoint cycles such that each of them contains at least two vertices of W.

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order. In this paper, answering a question of Ng and Schultz, we give a sharp bound for the minimum degree guaranteeing that a graph is a $k$-ordered Hamiltonian graph under some mild restrictions. More precisely, we show that there are $\varepsilon, n_0> 0$ such that if $G$ is a graph of order $n\geq n_0$ with minimum degree at least $\lceil \frac{n}{2} \rceil + \lfloor \frac{k}{2} \rfloor - 1$ and $2\leq k \leq \eps n$, then $G$ is a $k$-ordered Hamiltonian graph. It is also shown that this bound is sharp for every $2\leq k \leq \lfloor \frac{n}{2} \rfloor$.

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.

Cycles and stability

Let k and n be two integers such that k ≥ 0 and n ≥ 3(k + 1). Let G be a graph of order n with minimum degree at least ⌈(n + k)/2⌉. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. © 1995, John Wiley & Sons, Inc.

Bounds on the paired domination number of graphs with minimum degree at least three

A proof of a conjecture on the differential of a subcubic graph

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ⩾ 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ⩾ 2δ2, then for any 3 ⩽ l1 ⩽ l2 ⩽ ⃛ ⩽ lk ⩽ n, 1 = k = [(δ ‐ 1)/2], such graph has K edge disjoint cycles with lengths l1, l2…lk, respectively. In particular, when l1 = l2 = ⃛ = lk = n and k = [(δ ‐ 1)/2], the graph contains [(δ ‐ 1)/2] edge disjoint hamiltonian cycles.