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.

Chromatic-index-critical graphs of orders 13 and 14

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

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).

The k-dominating graph $$D_k(G)$$ of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to $$D_k(G)$$ for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order $$n\ge 2$$ and with $$G\cong D_k(G)$$ , then $$k=2$$ and $$G \cong K_{1,n-1}$$ for some $$n\ge 4$$ . It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, $$C_6$$ and $$C_8$$ are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.

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.

On Finite s-Transitive Graphs of Odd Order

Irregularity strength of dense graphs

On closed distance magic circulants of valency up to 5

The complete twisted graph of order n, denoted by $$T_n$$ , is a complete simple topological graph with vertices $$u_1, u_2, \ldots , u_n$$ such that two edges $$u_iu_j$$ and $$u_{i'}u_{j'}$$ cross if and only if $$ i<i'<j'<j $$ or $$ i'<i<j<j' $$ . The convex geometric complete graph of order n, denoted by $$G_n$$ , is a convex geometric graph with vertices $$v_1, v_2, \ldots , v_n$$ placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set $$\{v_1,v_2,\ldots , v_n \}$$ having the property of being planar when embedded in both $$T_n$$ and $$G_n$$ . Given a connected graph G the (combinatorial) tree graph $$\mathcal {T}(G)$$ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in $$\mathcal {T}(G)$$ if there are edges $$e \in P$$ and $$f\in Q$$ such that $$Q=P - e + f$$ . For all positive integers n, we denote by $$\mathcal {T}(n)$$ the graph $$\mathcal {T}(K_n)$$ . The biplanar tree graph, $$\mathcal {B}(n)$$ , is the subgraph of $$\mathcal {T}(n)$$ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.