On the Structure of Dominating Graphs

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.

Binding numbers and [formula omitted]-factors excluding a given k-factor

On fractional [formula omitted]-critical graphs

Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with $\(p\geq\frac{(a+b-1)(a+b-2)+bn-2}{a}\)$ and f be an integer-valued function defined on V(G) such that a ≤ f(x) ≤ b for all x ∈ V(G). Let h: E(G) → [0, 1] be a function. If ∑e∋xh(e) = f(x) holds for any x ∈ V(G), then we call G[Fh] a fractional f-factor of G with indicator function h, where Fh = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. In this paper, it is proved that G is a fractional (f, n)-critical graph if $\(|N_G(X)|>\frac{(b-1)p+|X|+bn-1}{a+b-1}\)$ for every non-empty independent subset X of V(G), and $\(\delta(G)>\frac{(b-1)p+a+b+bn-2}{a+b-1}\)$. Furthermore, it is shown that the result in this paper is best possible in some sense.

formula omitted]-factors in graphs

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved $$m{\left( {n,1} \right)} = \frac{{n^{2} }}{4}$$ for even n and $$m{\left( {n,2} \right)} = {\left\lfloor {\frac{{n{\left( {n + 1} \right)}}}{4}} \right\rfloor }$$, respectively. Recently, Johann confirmed the following conjectures of Hendry: $$m{\left( {n,k} \right)} = \frac{{nk}}{2} + {\left( {{}^{{n - k}}_{2} } \right)}$$ for $$k > \frac{n}{2}$$ and kn even and $$m{\left( {n,k} \right)} = \frac{{n^{2} }}{4} + {\left( {k - 1} \right)}\frac{n}{4}$$ for n = 2kq, where q is a positive integer. In this paper we prove $$m{\left( {n,k} \right)} = k^{2} + {\left( {{}^{{n - k}}_{2} } \right)}$$ for $$\frac{n}{3} \leqslant k < \frac{n}{2}$$ and kn even, and we determine m(n, 3).

A Ramsey-type Theorem for Multiple Disjoint Copies of Induced Subgraphs (Extended Abstract)

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 p be a hereditary graph property. The p‐chromatic number of a graph is the minimal number of classes in a vertex partition wherein each class spans a subgraph with property p. For the property p of edgeless graphs the p‐chromatic number is just the usual chromatic number, whose value is known to be (1/2 + o(1))n/log2 n for almost every graph of order n. We show that we may associate with every nontrivial hereditary property p an explicitly defined natural number r = r(p), and that the p‐chromatic number is then (l/2r + o(1))n/log2 n for almost every graph of order n. © 1995 John Wiley &amp; Sons, Inc.

Let H be a simple and connected graph. A shackle of graph H, denoted by G = shack(H, v, n), is a graph G constructed by non-trivial graphs H1, H2, …, Hn such that, for every 1 ≤ s, t ≤ n, Hs and Ht have no a common vertex with |s − t| ≥ 2 and for every 1 ≤ i ≤ n − 1, Hi and Hi+1 share exactly one common vertex v, called connecting vertex, and those k − 1 connecting vertices are all distinct. The graph G is said to be an (a*, d*)-H-antimagic total graph of second order if there exist a bijective function f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights form an arithmetic sequence of second order of , where a* and d* are positive integers and n is the number of all subgraphs isomorphic to H. An (a*, d*)-H-antimagic total labeling of second order f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a*, d*)-H antimagic total labeling of second order of G = shack(H, v, n) by using a partition technique of second order.

This paper introduces the concept of upper detour monophonic domination number of a graph. For a connected graph G with vertex set V (G), a set M ⊆ V (G) is called minimal detour monophonic dominating set, if no proper subset of M is a detour monophonic dominating set. The maximum cardinality among all minimal monophonic dominating sets is called upper detour monophonic domination number and is denoted by γ+ dm (G). For any two positive integers p and q with 2 ≤ p ≤ q there is a connected graph G with γm (G) = γdm (G) = p and γ+ dm (G) = q. For any three positive integers p, q, r with 2 < p < q < r, there is a connected graph G with m(G) = p, γdm (G) = q and γ+ dm (G) = r. Let p and q be two positive integers with 2 < p < q such that γdm (G) = p and γ+ dm (G) = q. Then there is a minimal DMD set whose cardinality lies between p and q. Let p, q and r be any three positive integers with 2 ≤ p ≤ q ≤ r. Then, there exist a connected graph G such that γm (G)= p, γ+ dm (G) = q and |V (G)| = r.

On Finite s-Transitive Graphs of Odd Order

Uniquely colorable graphs

For a graph G, δ(G) denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x) + d(y) ≥ n for each pair of non-adjacent vertices x, y in G ,t henG is pancyclic or G = Kn/2,n/2. In 2001, Xu proved that, if G is a 2-connected graph of order n ≥ 6a nd|N (x) ∪ N (y) |+ δ(G) ≥ n for each pair of non-adjacent vertices x, y in G ,t henG is pancyclic or G = Kn/2,n/2. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n ≥ 6a nd |N (x) ∪ N (y) |+ d(w) ≥ n for any three vertices x, y ,w of d(x, y) = 2a ndwx or wyE(G) in G ,t henG is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results. We generalize two well-known degree sum and neighborhood union for characterizing Hamiltonian graphs, in particular for vertex-pancyclic. We first give a few definitions and some notation. We consider only finite undirected graphs with no loops or multiples. We denote by δ(G) the minimum degree of G, and Kh the complete graph of order h .I fu is a vertex and H is a subgraph of G, then let NH (u) ={ v ∈ V (H ) : uv ∈ E(G)} be the vertex set of H that are adjacent to vertex u, and if S is a vertex set or subgraph of G, then set NH (S) =∪ u∈S NH (u) and set N (u )= N (u) ∪{ u} .L etG − H and G(S) denote the subgraphs of G induced by V (G) − V (H ) and S, respectively. If Cm = x1x2 ··· xm x1 is a cycle of order m ,l etN + Cm (u) ={ xi+1 : xi ∈ NCm (u)}, N − Cm (u) = {xi−1 : xi ∈ NCm (u)}, and N ± Cm (u) = N +

Irregularity strength of dense graphs