Path Pn Research Articles

On the Maximum Number of Spanning Copies of an Orientation in a Tournament

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=Cn, the directed Hamilton cycle, T(Cn) ≥ (e−o(1))n!/2n, and it was observed by Alon that already R(Cn) ≥ (e−o(1))n!/2n. Similar results hold for the directed Hamilton path Pn. In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (ek−o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (ek−o(1))n!/2e(H).

On The Partition Dimension of Cm + Pn Graph

Let G be a connected graph with V (G) = {v1, v2, …, vi} and E(G) = {e1, e2, …, ej}, where V (G) is vertex set and E(G) is edge set. If S ⊆ V (G) and v ∈ V (G), then the distance between v and S is de ned by d(v, S) = min{d(v, x)|x ∈ S}. For an ordered k-partition Π = {S1, S2, …, Sk} of V (G), the representation of v with respect to Π is r(v|Π) with r(v|Π) = (d(v, S1), d(v, S2), …, d(v, Sk)). If the representation of v ∈ V (G) with respect to Π are distinct, so Π is called a resolving partition of V (G). The minimum cardinality of resolving partition Π is called a partition dimension of G, denoted by pd(G). In this paper, we study the partition dimension of a Cm + Pn graph. Cm + Pn graph is a graph formed from join operation of cycle graph Cm with order m ≥ 3 and path Pn with order n ≥ 2. Cm + Pn is the union Cm ∪ Pn together with all edges uavb, for ua ∈ V (Cm) and vb ∈ V (Pn) with 1 ≤ a ≤ m and 1 ≤ b ≤ n. We obtain the partition dimension of Cm + Pn graph is pd(C3 + Pn) = g where g is the smallest positive integer such that n ≤ 5g − 12 for g = 5 and for g ≥ 6, and pd(Cq + Pn) = min{p + f, r + t, x + y} for q ≥ 4 and n ≥ 2 where p, f, r, t, x and y are some positive integers related to the number of partition classes containing vertices of Cq and Pn.

On The Partition Dimension of a Lollipop Graph and a Generalized Jahangir Graph

Let G be a connected graph with vertex set V (G), such that V (G) can be divided into any partition set S. The set Π with S ∈ Π is a resolving partition of G if each vertex in G has a distinct representation with respect to Π, and Π is an ordered k−partition. The minimum cardinality of resolving k−partitions of V (G) is called a partition dimension of G, denoted by pd(G). The lollipop graph Lm,n is a graph obtained by joining a complete graph Km to a path Pn with a bridge. A generalized Jahangir graph is a graph consisting of a cycle Cmn and one additional vertex which is adjacent to n vertices of Cmn at m distance to each other on Cmn. Many researchers have conducted research in determining the partition dimension for speci c graph classes. They are as references to determine some of the graph classes that haven’t been studied previously.In this paper, we determine the partition dimension of a lollipop graph Lm,n and a generalized Jahangir graph Jm,n. The research methods in this paper is a book study.The results of this paper are as follows. We obtain the partition dimension of a lollipop graph pd(Lm,n) = m for m ≥ 3 and n ≥ 1. The partition dimension of a generalized Jahangir graph consists of two cases. We showed that pd(Jm,n) = 3 for n = 3, 4, 5 and we prove that .

Variants of the Erdős–Szekeres and Erdős–Hajnal Ramsey problems

Given integers ℓ,n, the ℓth power of the path Pn is the ordered graph Pnℓ with vertex set v1<v2<⋯<vn and all edges of the form vivj where |i−j|≤ℓ. The Ramsey number r(Pnℓ,Pnℓ) is the minimum N such that every 2-coloring of [N]2 results in a monochromatic copy of Pnℓ. It is well-known that r(Pn1,Pn1)=(n−1)2+1. For ℓ>1, Balko–Cibulka–Král–Kynčl proved that r(Pnℓ,Pnℓ)<cℓn128ℓ and asked for the growth rate for fixed ℓ. When ℓ=2, we improve this upper bound substantially by proving r(Pn2,Pn2)<cn19.5. Using this result, we determine the correct tower growth rate of the k-uniform hypergraph Ramsey number of a (k+1)-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.

Kite graphs determined by their spectra

A kite graph Kin, ω is a graph obtained from a clique Kω and a path Pn−ω by adding an edge between a vertex from the clique and an endpoint from the path. In this note, we prove that Kin,n−1 is determined by its signless Laplacian spectrum when n ≠ 5 and n ≥ 4, and Kin,n−1 is also determined by its distance spectrum when n ≥ 4.

(a, d)-Distance Antimagic Labeling of Some Types of Graphs

We analyze the necessary existence conditions for (a, d)-distance antimagic labeling of a graph G = (V, E) of order n. We obtain theorems that expand the family of not (a, d) -distance antimagic graphs. In particular, we prove that the crown Pn ? P1 does not admit an (a, 1)-distance antimagic labeling for n ? 2 if a ? 2. We determine the values of a at which path Pn can be an (a, 1)-distance antimagic graph. Among regular graphs, we investigate the case of a circulant graph.

Extremal H-colorings of trees and 2-connected graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area.First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H-colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n-vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs.

Further results regarding the sum of domination number and average eccentricity

The average eccentricity of a graph G, denoted by ecc(G), is the mean value of eccentricities of all vertices of G. Let Dn, i be the n-vertex tree obtained from a path Pn−1=v1v2⋯vn−1 by attaching a pendent vertex to vi. In [13], it was shown that the maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs is attained by Dn, 3 when n≡0(mod3), and attained by the path Pn when n¬≡0(mod3). In this paper, we will further determine the second maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs. It is interesting that the graphs attaining that second maximum value have three cases, which is Dn, 6 when n≡0(mod3),Dn, 3 when n≡1(mod3), and Tn when n≡2(mod3), where Tn is the n-vertex tree obtained from a path Pn−2=v1v2⋯vn−2 by attaching a pendent vertex to v3, and a pendent vertex to vn−4.

On the extremal values of general degree-based graph entropies

In this note we prove one part of the conjecture about upper and lower bounds of the degree-based graph entropy Ik(T) in the class of trees introduced in [S. Cao, M. Dehmer, Y. Shi, Extremality of degree-based graph entropies, Information Sciences 278 (2014) 22–33.] using Lagrange multipliers and Jensen’s inequality, and disprove the other part by providing a family of counter-examples. Our main result is the following: the path Pn is unique tree on n vertices that maximizes Ik(T) for k > 0, and the star Sn is unique tree on n vertices that minimizes Ik(T) for k ≥ 1.

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G,H) is the smallest N such that every red–blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G,H)≥(|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)-coloring of H. A graph G is called H-good if R(G,H)=(|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n≥4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan.

Square Difference Labeling of Some Graphs

The paper considers some methods to construct square difference labeling of cactus cycle Cm(n) of single-point connection of n copies of cycle Cm and n copies of path P2 of single-point connection of n copies of cycle Cm and path Pn+1 as well as disjoint union of single-point connection of n copies of cycle Cm with path Pn.

Game total domination for cycles and paths

In this paper, we continue the study of the recently introduced total domination game in graphs. A vertex u in a graph G totally dominates a vertex v if u is adjacent to v in G. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game played on G consists of two players, named Dominator and Staller, who alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. Dominator’s goal is to totally dominate the graph as fast as possible, and Staller wishes to delay the process as much as possible. The game total domination number, γtg(G), of G is the number of vertices chosen when Dominator starts the game and both players play optimally. The Staller-start game total domination number, γtg′(G), of G is the number of vertices chosen when Staller starts the game and both players play optimally. In this paper we determine γtg(G) and γtg′(G) when G is a cycle or a path. In particular, we show that for a cycle Cn on n≥3 vertices, γtg(Cn)=⌊2n+13⌋−1 when n≡4(mod6) and γtg(Cn)=⌊2n+13⌋ otherwise. Further, γtg′(Cn)=⌊2n3⌋−1 when n≡2(mod6) and γtg′(Cn)=⌊2n3⌋ otherwise. For a path Pn on n≥3 vertices, we show that γtg′(Pn)=⌈2n3⌉. Further, γtg(Pn)=⌊2n3⌋ when n≡5(mod6) and ⌈2n3⌉ otherwise.

Resolvent Estrada index of cycles and paths

Let G be a simple graph of order n. The resolvent Estrada index of G is defined as EEr = nΣi=1 (1- λi/(n-1)^-1, where λ1;λ2;...;λn are the eigenvalues of G. Formulas for computing EEr of the cycle Cn and the path Pn are derived. The precision of these approximations are shown to be excellent. We also examine the difference and relations between the Estrada index and the resolvent Estrada index of Cn and Pn.

On the distance spectrum of trees

Let G be a connected graph with vertex set V(G) = {v1, v2,..., vng} and edge set E(G).D(G) = (dij)nxn is the distance matrix of G, where dij denotes the distance between vi and vj. Let ?1(D) ? ?2(D)?... ? ?n(D) be the distance spectrum of G. A graph G is said to be determined by its distance spectrum if any graph having the same distance spectrum as G is isomorphic to G. Trees can not be determined by its distance spectrum. Naturally, we prove that two kinds of special trees path Pn and double star S(a,b) are determined by their distance spectra in this paper.

Game chromatic number of lexicographic product graphs

In this paper, we determine the exact values of the game chromatic number of lexicographic product of path P2 with path Pn, star K1,n and wheel Wn. Also we give an upper bound for the game chromatic number of lexicographic product of any two simple graphs G and H.

Bounds on locating total domination number of the Cartesian product of cycles and paths

The problem of placing monitoring devices in a system in such a way that every site in the safeguard system (including the monitors themselves) is adjacent to a monitor site can be modeled by total domination in graphs. Locating-total dominating sets are of interest when the intruder/fault at a vertex precludes its detection in that location. A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V−S are totally dominated by distinct subsets of the total dominating set. The locating-total domination number of a graph G is the minimum cardinality of a locating-total dominating set of G. In this paper, we study the bounds on locating-total domination numbers of the Cartesian product Cm□Pn of cycles Cm and paths Pn. Exact values for the locating-total domination number of the Cartesian product C3□Pn are found, and it is shown that for the locating-total domination number of the Cartesian product C4□Pn this number is between ⌈3n2⌉ and ⌈3n2⌉+1 with two sharp bounds.

Further Results on Vertex Equitable Graphs

Let G be a graph with p vertices and q edges and A={0,1,2,…,⌈q2⌉}. A graph G is said to be vertex equitable if there exists a vertex labeling f:V(G)→A that induces an edge labeling f⁎ defined by f⁎(uv)=f(u)+f(v) for all edges uv such that for all a and b in A, |vf(a)−vf(b)|≤1 and the induced edge labels are 1,2,3,…,q where vf(a) is the number of vertices v with f(v)=a for a∈A. The vertex labeling f is known as vertex equitable labeling. In this paper, we prove that NQ(m), DA(Qm)⊙nK1, DA(Tm)⊙nK1, the graph obtained by subdividing the edges of the path Pn in Pn⊙K1 and Pn⊙2K1 are vertex equitable graphs.

GRACEFUL LABELING OF EUCALYPTUS CLOUD CONNECTED IN A PATH

Cloud computing swears on divvying of resources to reach coher- ence and economies of exfoliation similar to a utility (like the electricity grid) over a network. Eucalyptus produces open source software for edifice AWS - simpatico individual and loanblend clouds. A method for assigning IP address for each node of Eucalyptus cloud is proposed using graceful labeling algorithm. A graceful labeling of n copies of Eucalyptus cloud structure tree connected in a path Pn is shown. It is proved that IP addresses generated by the proposed algorithm is unique in any network.

The minimum span of [formula omitted]-labelings of generalized flowers

Given a positive integer d, an L(d,1)-labeling of a graph G is an assignment of nonnegative integers to its vertices such that adjacent vertices must receive integers at least d apart, and vertices at distance two must receive integers at least one apart. The λd-number of G is the minimum k so that G has an L(d,1)-labeling using labels in {0, 1, …, k}. Informally, an amalgamation of two disjoint graphs G1 and G2 along a fixed graph G0 is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G0, one in G1 and the other in G2. A flower is an amalgamation of two or more cycles along a single vertex. We provide the exact λ2-number of a generalized flower which is the Cartesian product of a path Pn and a flower, or equivalently, an amalgamation of cylindrical rectangular grids along a certain Pn. In the process, we provide general upper bounds for the λd-number of the Cartesian product of Pn and any graph G, using circularL(d+1,1)-labelings of G where the labels {0, 1, …, k} are arranged sequentially in a circle and the distance between two labels is the shortest distance on the circle.

Solitaire clobber on circulant graphs

Solitaire Clobber is a one-player combinatorial game on graphs. Each vertex of a graph G starts with a black or a white stone. A stone on one vertex can clobber an adjacent stone of the opposite color, removing it and taking its place. The goal is to minimize the number of stones remaining when no further move is possible. An initial configuration is k-reducible if it can be reduced to k stones. A graph is strongly 1-reducible if, for any vertex v, any initial configuration that is not monochromatic outside v can be reduced to one stone, on v, of either color. Every such graph has a Hamiltonian path ending at v. For the path Pn, we prove that the rth distance power Pnr is strongly 1-reducible when r≥3 but not when r=2 (Pn2 is 2-reducible). As a consequence, circulant graphs containing edges of lengths 1, 2, and 3 are strongly 1-reducible; we show also that those containing Cn2 are 1-reducible.