Some Distance Antimagic Labeled Graphs

Let be a graph of order n. Let be a bijection. For any vertex , the neighbor sum is called the weight of the vertex and is denoted by where N(v) is the open neighborhood of If for any two distinct vertices and then f is called a distance antimagic labelling. If the graph G admits such a labelling, then G is said to be a distance antimagic graph. This study gives a distance antimagic labelling for tensor product of two complete graph and sufficient condition so that the tensor product of a regular graph and complete graph is a distance antimagic graph.

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

A handicap distance antimagic labeling of a graph G = ( V , E ) with n vertices is a bijection f : V → {1, 2, …, n } with the property that f ( x i ) = i , the weight w ( x i ) is the sum of labels of all neighbors of x i , and the sequence of the weights w ( x 1 ), w ( x 2 ), …, w ( x n ) forms an increasing arithmetic progression. A graph G is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling. We construct r -regular handicap distance antimagic graphs of order $n \equiv 0 \pmod{8}$ for all feasible values of r .

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.

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 & Sons, Inc.

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.

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 +

An Ore-type condition for the existence of two disjoint cycles

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.

Minimum degree condition for proper connection number 2

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) &gt; 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 $\(|N_G(X)|&gt;\frac{(b-1)p+|X|+bn-1}{a+b-1}\)$ for every non-empty independent subset X of V(G), and $\(\delta(G)&gt;\frac{(b-1)p+a+b+bn-2}{a+b-1}\)$. 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).