Discrete Mathematics and Combinatorics Commons^™

Graham's Pebbling Conjecture Holds For The Product Of A Graph And A Sufficiently Large Complete Graph, Nopparat Pleanmani

Theory & Applications of Graphs

For connected graphs G and H, Graham conjectured that π(G □ H) ≤ π(G) π(H) where π(G), π (H), and π(G □ H) are the pebbling numbers of G, H, and the Cartesian product G □ H, respectively. In this paper, we show that the inequality holds when H is a complete graph of sufficiently large order in terms of graph parameters of G.

Rearrangement Operations On Unrooted Phylogenetic Networks, Remie Janssen, Jonathan Klawitter

Theory & Applications of Graphs

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including …

Conditional Strong Matching Preclusion Of The Alternating Group Graph, Mohamad Abdallah, Eddie Cheng

Theory & Applications of Graphs

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the n-dimensional alternating group graph AG_n.

Forcibly-Biconnected Graphical Degree Sequences: Decision Algorithms And Enumerative Results, Kai Wang

Theory & Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n. Based on these enumerative …

Laplacian Spectral Characterization Of Signed Sun Graphs, Fatemeh Motialah, Mohammad Hassan Shirdareh Haghighi

Theory & Applications of Graphs

A sun SG_n is a graph of order 2n consisting of a cycle C_n, n ≥ 3, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if n is odd.

Fractional Strong Matching Preclusion For Two Variants Of Hypercubes, Huifen Ge, Tianlong Ma, Miaolin Wu, Yuzhi Xiao

Theory & Applications of Graphs

Let F be a subset of edges and vertices of a graph G. If G-F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each.

Colored Complete Hypergraphs Containing No Rainbow Berge Triangles, Colton Magnant

Theory & Applications of Graphs

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs.

Maximum Oriented Forcing Number For Complete Graphs, Yair Caro, Ryan Pepper

Theory & Applications of Graphs

The maximum oriented k-forcing number of a simple graph G, written MOF_k(G), is the maximum directed k-forcing number among all orientations of G. This invariant was recently introduced by Caro, Davila and Pepper in [6], and in the current paper we study the special case where G is the complete graph with order n, denoted K_n . While MOF_k(G) is an invariant for the underlying simple graph G, MOF_k(K_n) can also be interpreted as an interesting property for tournaments. Our main results …

Matching Preclusion Of The Generalized Petersen Graph, Ajay Arora, Eddie Cheng, Christopher Melekian

Theory & Applications of Graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. In this paper we determine the matching preclusion number for the generalized Petersen graph P(n,k) and classify the optimal sets.

Series-Parallel Operations With Alpha-Graphs, Christian Barrientos, Sarah Minion

Theory & Applications of Graphs

Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path Ρ₁₁}, are α-graphs. We also show that the one-point union of up to four copies of Κ_n,n is an …

Fractional Matching Preclusion For Butterfly Derived Networks, Xia Wang, Tianlong Ma, Chengfu Ye, Yuzhi Xiao, Fang Wang

Theory & Applications of Graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [18] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G, denoted by fmp(G), is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G, denoted by fsmp(G), is the minimum number of vertices and edges whose deletion …

Singular Ramsey And Turán Numbers, Yair Caro, Zsolt Tuza

Theory & Applications of Graphs

We say that a subgraph F of a graph G is singular if the degrees d_G(v) are all equal or all distinct for the vertices v ∈ V (F). The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m at least n, in every edge 2-coloring of K_m, at least one of the color classes contains F as a singular subgraph. In a similar flavor, the singular Turán number Ts(n,F) is defined as the maximum number of edges in a graph of order n, …

Subsets Of Vertices Of The Same Size And The Same Maximum Distance, Maria Axenovich, Dominik Duerrschnabel

Theory & Applications of Graphs

For a simple connected graph G=(V,E) and a subset X of its vertices, let

d*(X) = max {dist}_G(x,y): x,y ∈ X}

and let h*(G) be the largest k such that there are disjoint vertex subsets A and B of G, each of size k such that d*(A) = d*(B). Let h*(n) = min {h*(G): |V(G)|=n}. We prove that h*(n) =⌊(n+1)/3 ⌋, for n ≥ 6. This solves the homometric set problem restricted to the largest distance exactly. In addition we compare h*(G) with a respective function h_diam(G), where …

Finite Simple Graphs And Their Associated Graph Lattices, James B. Hart, Brian Frazier

Theory & Applications of Graphs

In his 2005 dissertation, Antoine Vella explored combinatorical aspects of finite graphs utilizing a topological space whose open sets are intimately tied to the structure of the graph. In this paper, we go a step further and examine some aspects of the open set lattices induced by these topological spaces. In particular, we will characterize all lattices that constitute the opens for finite simple graphs endowed with this topology, explore the structure of these lattices, and show that these lattices contain information necessary to reconstruct the graph and its complement in several ways.

Reducing The Maximum Degree Of A Graph By Deleting Vertices: The Extremal Cases, Peter Borg, Kurt Fenech

Theory & Applications of Graphs

Let λ(G) denote the smallest number of vertices that can be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. In a recent paper, we proved that if n is the number of vertices of G,k is the maximum degree of G, and t is the number of vertices of degree k, then λ: (G) ≤ n+(k-1)t}{2k}. We also showed that λ (G) ≤ \frac{n}{k+1} if G is a tree. In this paper, we provide a new proof of the first bound and use it to determine the graphs that …

Minimal Graphs With A Specified Code Map Image, Paul Feit

Theory & Applications of Graphs

Let G be a graph and ^e1,…^en be n distinct vertices. Let ρ be the metric on G. The code map on vertices, corresponding to this list, is c(x)=(ρ (x,^e1),…ρ (x,^en)). This paper introduces a variation: begin with V ⊆ Ζ^n for some n, and consider assignments of edges E such that the identity function on V is a code map for G=(V,E). Refer to such a set E as a code-match.

An earlier paper classified subsets of V for which at least one code-match exists. We prove …

Integer-Antimagic Spectra Of Disjoint Unions Of Cycles, Wai Chee Shiu

Theory & Applications of Graphs

Let A be a non-trivial abelian group. A simple graph G = (V, E) is A-antimagic if there exists an edge labeling f: E(G) \to A \setminus \{0\} such that the induced vertex labeling f^+: V(G) \to A, defined by f^+(v) = \sum_{uv\in E(G)}f(uv), is injective. The integer-antimagic spectrum of a graph G is the set IAM(G) = \{k\;|\; G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic and } k \geq 2\}. In this paper, we determine the integer-antimagic spectra of disjoint unions of cycles.

An Efficient Algorithm To Test Forcibly-Connectedness Of Graphical Degree Sequences, Kai Wang

Theory & Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly k-connected or not for every fixed k ≥ 2. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al's classic algorithm to enumerate zero-free graphical degree sequences of length n and Barnes and Savage's classic algorithm to enumerate graphical partitions of even integer …

Finite Asymptotic Clusters Of Metric Spaces, Viktoriia Bilet, Oleksiy Dovgoshey

Theory & Applications of Graphs

Let (X, d) be an unbounded metric space and let \tilde r=(r_n)_{n\in\mathbb N} be a sequence of positive real numbers tending to infinity. A pretangent space \Omega_{\infty, \tilde r}^{X} to (X, d) at infinity is a limit of the rescaling sequence \left(X, \frac{1}{r_n}d\right). The set of all pretangent spaces \Omega_{\infty, \tilde r}^{X} is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph (G_{X, \tilde r}, \rho_{X}) whose maximal cliques coincide with \Omega_{\infty, \tilde r}^{X} and the weight \rho_{X} is defined by metrics on \Omega_{\infty, \tilde r}^{X}. We describe the structure …

Gamma-Realizability And Other Musings On Inverse Domination, John Asplund, Joe Chaffee, James M. Hammer Iii, Matt Noble

Theory & Applications of Graphs

We introduce and study γ-realizable sequences. For a finite, simple graph G containing no isolated vertices, I ⊆ V(G) is said to be an inverse dominating set if I dominates all of G and I is contained by the complement of some minimum dominating set D. Define a sequence of positive integers (x₁,…, x_n) to be γ-realizable if there exists a graph G having exactly n distinct minimum dominating sets D₁,… D_n where for each i ∈ {1,… n}, the minimum size of an inverse dominating set in V(G) …

Policy-Preferred Paths In As-Level Internet Topology Graphs, Mehmet Engin Tozal

Theory & Applications of Graphs

Using Autonomous System (AS) level Internet topology maps to determine accurate AS-level paths is essential for network diagnostics, performance optimization, security enforcement, business policy management and topology-aware application development. One significant drawback that we have observed in many studies is simplifying the AS-level topology map of the Internet to an undirected graph, and then using the hop distance as a means to find the shortest paths between the ASes. A less significant drawback is restricting the shortest paths to only valley-free paths. Both approaches usually inflate the number of paths between ASes; introduce erroneous paths that do not conform to …

Traveling In Networks With Blinking Nodes, Braxton Carrigan, James Hammer

Theory & Applications of Graphs

We say that a blinking node system modulo n is an ordered pair (G,L) where G is a graph and L is an on-labelling which indicates when vertices can be visited. An On-Hamiltonian walk is a sequence of all the vertices of G such that the position of each vertex modulo n is an integer of the label of that vertex. This paper will primarily investigate finding the shortest On-Hamiltonian walks in a blinking node system on complete graphs and complete bipartite graphs but also establishes the terminology and initial observations for working with blinking node systems on other graphs.

Generalized Matching Preclusion In Bipartite Graphs, Zachary Wheeler, Eddie Cheng, Dana Ferranti, Laszlo Liptak, Karthik Nataraj

Theory & Applications of Graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal such sets are precisely sets of edges incident to a single vertex. The conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond these, and it is defined as the minimum number of edges whose deletion results in a graph with neither isolated vertices nor perfect matchings. In this paper we generalize this concept to get a hierarchy of …

A General Lower Bound On Gallai-Ramsey Numbers For Non-Bipartite Graphs, Colton Magnant

Theory & Applications of Graphs

Given a graph H and a positive integer k, the k-color Gallai-Ramsey number gr_k(K3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph K_n contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs H to a value that we conjecture to …

On The Planarity Of Generalized Line Graphs, Khawlah H. Alhulwah, Mohra Zayed, Ping Zhang

Theory & Applications of Graphs

One of the most familiar derived graphs is the line graph. The line graph L(G) of a graph G is that graph whose vertices are the edges of G where two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. Two nontrivial paths P and Q in a graph G are said to be adjacent paths in G if P and Q have exactly one vertex in common and this vertex is an end-vertex of both P and Q. For an integer ℓ ≥ 2, the ℓ -line graph Lℓ (G) of a …

A Survey On Monochromatic Connections Of Graphs, Xueliang Li, Di Wu

Theory & Applications of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it.
In this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.

Edge Colorings Of Complete Multipartite Graphs Forbidding Rainbow Cycles, Peter Johnson, Andrew Owens

Theory & Applications of Graphs

It is well known that if the edges of a finite simple connected graph on n vertices are colored so that no cycle is rainbow, then no more than n-1 colors can appear on the edges. In previous work it has been shown that the essentially different rainbow-cycle-forbidding edge colorings of K_n with n-1 colors appearing are in 1-1 correspondence with (can be encoded by) the (isomorphism classes of) full binary trees with n leafs. In the encoding, the natural Huffman labeling of each tree arising from the assignment of 1 to each leaf plays a role. Very recently …

On The Graceful Cartesian Product Of Alpha-Trees, Christian Barrientos, Sarah Minion

Theory & Applications of Graphs

A graceful labeling of a graph G of size n is an injective assignment of integers from the set {0,1,…,n} to the vertices of G such that when each edge has assigned a weight, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an α-labeling when the graph G is bipartite, with stable sets A and B, and the labels assigned to the vertices in A are smaller than the labels assigned to the vertices in B. In this …

Note On 6-Regular Graphs On The Klein Bottle, Michiko Kasai, Naoki Matsumoto, Atsuhiro Nakamoto, Takayuki Nozawa, Hiroki Seno, Yosuke Takiguchi

Theory & Applications of Graphs

Altshuler classified six regular graphs on the torus, but Thomassen and Negami gave different classifications for six regular graphs on the Klein bottle. In this note, we unify those two classifications, pointing out their difference and similarity.

An Updated Survey On Rainbow Connections Of Graphs - A Dynamic Survey, Xueliang Li, Yuefang Sun

Theory & Applications of Graphs

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the …