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Let S be a set of at least two vertices in a graph G. A subtree T of G is an S -Steiner tree if S ⊆ V (T). Two S -Steiner trees T 1 and T 2 are internally disjoint (resp. edge-disjoint) if E (T 1) ∩ E (T 2) = 0̸ and V (T 1) ∩ V (T 2) = S (resp. E (T 1) ∩ E (T 2) = 0̸). Let κ G (S) (resp. λ G (S)) be the maximum number of internally disjoint (resp. edge-disjoint) S -Steiner trees in G , and let κ k (G) (resp. λ k (G)) be the minimum κ G (S) (resp. λ G (S)) for S ranges over all k -subsets of V (G). It is well-known that the connectivity of the line graph of a graph G is closely related to the edge-connectivity of G. Chartrand et al., Li et al., and Li et al. showed that κ k (L (G)) ≥ λ k (G) for k = 2 , 3 , 4 , respectively. In [Discrete Math. 341(2018), 1945–1951.], Li et al. also proposed the following conjecture: κ k (L (G)) ≥ λ k (G) for integer k ≥ 2 and a graph G with at least k vertices. In this paper, we show that κ k (L (G)) ≥ λ k (G) − m a d (G) 2 2 for general k , where m a d (G) is the maximum average degree of a graph G. We also show that κ k (L (G)) ≥ λ k (G) − r 2 2 for any integers k and r such that k ≤ r 2 . Finally, we show that the conjecture holds for k = 5 , and the conjecture holds for the wheel graph and the complete bipartite graph K 3 , n. [ABSTRACT FROM AUTHOR]

Considers a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries. Definition of the string graph problem; Characterization of map graphs in terms of planar bipartite graphs; Ways in which the cliques in map graphs are realized.

Abstract Graph energy can be extended to digraphs via the trace norm. The trace norm of a digraph is the trace norm of its adjacency matrix, i.e. the sum of its singular values. In this work we find the oriented graphs that attain minimal and maximal trace norm over the set of oriented bicyclic graphs. [ABSTRACT FROM AUTHOR]

In this article, we propose a polynomial‐time algorithm to test whether a given graph contains a subdivision of K4 as an induced subgraph. This continues the study of detecting an induced subdivision of H for some fixed graph H, which is still far from being complete. Our result answers a question by Chudnovsky et al. and Lévêque et al. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to construct pliable fractional repetition (FR) codes that are locally recoverable for distributed storage systems (DSSs). FR codes are integral in constructing a class of distributed storage codes with exact repair by transfer. Pliable FR codes are a new type of FR codes in which both the per-node storage and repetition degree can easily be adjusted simultaneously; thus, pliable FR codes are vital for DSSs in which parameters can dynamically change over time. However, the constructions of pliable FR codes with repair locality remain unknown. In addition, the tradeoffs between the code minimum distance of an FR code and its repair locality are not fully understood. To address these problems, this paper first presents general results regarding FR codes. Subsequently, this paper presents an improved Singleton-like bound for locally recoverable FR codes under an additional requirement that each node must be part of a local structure that, upon failure, allows it to be exactly recovered by a simple download process. Moreover, this paper proposes a construction of locally recoverable FR codes that can achieve the proposed Singleton-like bound; this construction is based on bipartite graphs with a given girth. In particular, this paper also proposes a general bipartite-graph-based approach to constructing optimal pliable FR codes with and without repair localities; in this approach, a new family of bipartite graphs, called matching-feasible graphs, is introduced. Finally, this paper proposes the explicit constructions of optimal pliable FR codes by using a family of matching-feasible graphs with arbitrary large girth. Notably, in addition to attaining a Singleton-like bound for FR codes, the explicit pliable FR codes are optimal locally recoverable FR codes from two perspectives of repair locality. The explicit pliable FR codes can also be used as FR batch codes to provide load balancing in DSSs. [ABSTRACT FROM AUTHOR]

In this article, we propose a polynomial‐time algorithm to test whether a given graph contains a subdivision of K4 as an induced subgraph. This continues the study of detecting an induced subdivision of H for some fixed graph H, which is still far from being complete. Our result answers a question by Chudnovsky et al. and Lévêque et al. [ABSTRACT FROM AUTHOR]

For a graph G = (V,E), a double Roman dominating function (or just DRDF) is a function f: V → {0, 1, 2, 3} having the property that if f(v) = 0 for a vertex v, then v has at least two neighbors assigned 2 under f or one neighbor assigned 3 under f, and if f(v) = 1, then vertex v must have at least one neighbor w with f(w) = 2. The weight of a DRDF f is the sum f(V) = ΣvεV f(v), and the minimum weight of a DRDF on G is the double Roman domination number of G, denoted by dR(G). In this paper, we derive sharp upper and lower bounds on γdR(G) +γdR(G) and also γdR(G)γdR(G), where G is the complement of graph G. We also show that the decision problem for the double Roman domination number is NP- complete even when restricted to bipartite graphs and chordal graphs. [ABSTRACT FROM AUTHOR]

Abstract A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph G is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of G. We show a similar combinatorial interpretation for principal minors of signless Laplacian Q. We also prove that the number of odd cycles in G is less than or equal to det ⁡ (Q) 4 , where the equality holds if and only if G is a bipartite graph or an odd-unicyclic graph. [ABSTRACT FROM AUTHOR]

Reed and Seymour [1998] asked whether every graph has a partition into induced connected nonempty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is "no." In fact, we show that the answer is still "no" for several relaxations of the question. [ABSTRACT FROM AUTHOR]

We determine the structure of {C3,C5}‐free graphs with n vertices and minimum degree larger than n/5: such graphs are homomorphic to the graph obtained from a (5k−3)‐cycle by adding all chords of length 1(mod5), for some k. This answers a question of Messuti and Schacht. We deduce that the homomorphism threshold of {C3,C5}‐free graphs is 1/5, thus answering a question of Oberkampf and Schacht. [ABSTRACT FROM AUTHOR]

Let H be a set of connected graphs, each of which has order at least three, and suppose that there exist infinitely many connected H‐free graphs of minimum degree at least  two and all except for finitely many of them have a 2‐factor. In [J. Graph Theory, 64 (2010), 250–266], we proved that if |H|≤3, then one of the members in H is a star. In this article, we determine the remaining members of H and hence give a complete characterization of the pairs and triples of forbidden subgraphs. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Stiebitz 13 asserts that any graph with minimum degree at least s+t+1 can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This resolved a conjecture of Thomassen. In this article, we prove that for s,t≥2, if a graph G contains no cycle of length four and has minimum degree at least s+t−1, then G can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This improves the result of Diwan in 5, where he proved the same statement for graphs of girth at least five. Our proof also works for the case of variable functions, in which the bounds are sharp as showing by some polarity graphs. [ABSTRACT FROM AUTHOR]

The diameter of a graph gives the length of the longest path among all the shortest paths between any two vertices of the graph, and the diameter vulnerability problem measures the change in the diameter upon the deletion of edges. In this paper we determine the diameter vulnerability of the generalized Petersen graph GP[tk, k], for integers t ≥ 2 and k ≥ 1, and show that (except for some small cases) the diameter remains unchanged upon the deletion of one edge. This work contributes towards a solution of the well-known (∆, D, D', s)-problem, which attempts to find large graphs with maximum degree ∆ and diameter D such that the subgraphs obtained by deleting any set of up to s edges have diameter at most D ' , preferably equal to D itself. In cases when the delay in communication across a network is directly related to the length of the paths between stations, network designers generally prefer to opt for graphs having the property of being resistant to drastic shocks upon the deletion of edges. This reliability property makes this class of graphs ideal to be used for modeling interconnection networks. [ABSTRACT FROM AUTHOR]

A bipartite graph H is said to have Sidorenko's property if the probability that the uniform random mapping from V(H) to the vertex set of any graph G is a homomorphism is at least the product over all edges in H of the probability that the edge is mapped to an edge of G. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if H has Sidorenko's property, then the Cartesian product of H with an even cycle also has Sidorenko's property. [ABSTRACT FROM AUTHOR]

We consider first order expressible properties of random perfect graphs. That is, we pick a graph Gn uniformly at random from all (labeled) perfect graphs on n vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that Gn satisfies it does not converge as n → ∞. [ABSTRACT FROM AUTHOR]

A classic result of Erdős, Gyárfás and Pyber states that for every coloring of the edges of Kn with r colors, there is a cover of its vertex set by at most f(r)=O(r2logr) vertex‐disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of Kn but only on the number of colors. We initiate the study of this phenomenon in the case where Kn is replaced by the random graph G(n,p). Given a fixed integer r and p=p(n)≥n−1/r+ε, we show that with high probability the random graph G∼G(n,p) has the property that for every r‐coloring of the edges of G, there is a collection of f′(r)=O(r8logr) monochromatic cycles covering all the vertices of G. Our bound on p is close to optimal in the following sense: if p≪(logn/n)1/r, then with high probability there are colorings of G∼G(n,p) such that the number of monochromatic cycles needed to cover all vertices of G grows with n. [ABSTRACT FROM AUTHOR]

Let D(n,p) be the random directed graph on n vertices where each of the n(n−1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if p≥(logn+ω(1))/n then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically n!(p(1+o(1)))n. We also prove a hitting‐time version of this statement, showing that in the random directed graph process, as soon as every vertex has in‐/out‐degrees at least 1, there are typically n!(logn/n(1+o(1)))n directed Hamilton cycles. [ABSTRACT FROM AUTHOR]

We study two versions of graph coloring, where the goal is to assign a positive integer color to each vertex of an input graph such that adjacent vertices do not receive the same color assignment. For classic vertex coloring, the goal is to minimize the maximum color used, and for the sum coloring problem, the goal is to minimize the sum of colors assigned to all input vertices. In the offline variant, the entire graph is presented at once, and in online problems, one vertex is presented for coloring at each time, and the only information is the identity of its neighbors among previously known vertices. In batched graph coloring, vertices are presented in k batches, for a fixed integer k≥2, such that the vertices of a batch are presented as a set, and must be colored before the vertices of the next batch are presented. This last model is an intermediate model, which bridges between the two extreme scenarios of the online and offline models. We provide several results, including a general result for sum coloring and results for the classic graph coloring problem on restricted graph classes: We show tight bounds for any graph class containing trees as a subclass (forests, bipartite graphs, planar graphs, and perfect graphs, for example), and an interesting result for interval graphs and k=2, where the value of the (strict and asymptotic) competitive ratio depends on whether the graph is presented with its interval representation or not. [ABSTRACT FROM AUTHOR]

This article proposes the tolerance nearness measure (TNM) as a computationally reduced alternative to the graph edit distance (GED) for performing graph comparisons. The TNM is defined within the context of near set theory, where the central idea is that determining similarity between sets of disjoint objects is at once intuitive and practically applicable. The TNM between two graphs is produced using the Bron-Kerbosh maximal clique enumeration algorithm. The result is that the TNM approach is less computationally complex than the bipartite-based GED algorithm. The contribution of this paper is the application of TNM to the problem of quantifying the similarity of disjoint graphs and that the maximal clique enumeration-based TNM produces comparable results to the GED when applied to the problem of content-based image processing, which becomes important as the number of nodes in a graph increases. [ABSTRACT FROM AUTHOR]

An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for thedecomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph. [ABSTRACT FROM AUTHOR]

The neighborhood complex of a graph is the family of subsets of open neighborhoods of its vertices. The neighborhood polynomial is the ordinary generating function for the number of sets of the neighborhood complex with respect to their cardinality. This paper provides a new representation of the neighborhood polynomial as a sum over complete bipartite subgraphs of a graph. Using the close relation between the domination polynomial and the neighborhood polynomial of a graph, we can also give a new presentation of the domination polynomial. Finally we show that finding the number of certain double cliques of a graph is sufficient to determine the number of dominating sets of a graph. [ABSTRACT FROM AUTHOR]

Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns. [ABSTRACT FROM AUTHOR]

An octad triple is a set of three octads, octads being the blocks of the S(5, 8, 24) Steiner system. In this paper we determine the orbits of M24, the largest Mathieu group, upon the set of octad triples. [ABSTRACT FROM AUTHOR]

A graph G of order n≥3 is pancyclic if G contains a cycle of each possible length from 3 to n, and vertex pancyclic (edge pancyclic) if every vertex (edge) is contained on a cycle of each possible length from 3 to n. A chord of a cycle is an edge between two nonadjacent vertices of the cycle, and chorded cycle is a cycle containing at least one chord. We define a graph G of order n≥4 to be chorded pancyclic if G contains a chorded cycle of each possible length from 4 to n. In this article, we consider extensions of the property of being chorded pancyclic to chorded vertex pancyclic and chorded edge pancyclic. [ABSTRACT FROM AUTHOR]

A graph is called H-free if it has no induced subgraph isomorphic to H. A graph is called Ni-locally connected if G[{x∈V(G):1≤dG(w,x)≤i}] is connected and N2-locally connected if G[{uv:{uw,vw}⊆E(G)}] is connected for every vertex w of G, respectively. In this paper, we prove the following.Every 2-connected P7-free graph of minimum degree at least three other than the Petersen graph has a spanning Eulerian subgraph. This implies that every H-free 3-connected graph (or connected N4-locally connected graph of minimum degree at least three) other than the Petersen graph is supereulerian if and only if H is an induced subgraph of P7, where Pi is a path of i vertices.Every 2-edge-connected H-free graph other than {K2,2k+1:kis a positive integer} is supereulerian if and only if H is an induced subgraph of P4.If every connected H-free N3-locally connected graph other than the Petersen graph of minimum degree at least three is supereulerian, then H is an induced subgraph of P7 or T2,2,3, i.e., the graph obtained by identifying exactly one end vertex of P3,P3,P4, respectively.If every 3-connected H-free N2-locally connected graph is hamiltonian, then H is an induced subgraph of K1,4. We present an algorithm to find a collapsible subgraph of a graph with girth 4 whose idea is used to prove our first conclusion above. Finally, we propose that the reverse of the last two items would be true. [ABSTRACT FROM AUTHOR]

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. Bolloba´s and Scott proved that every regular graph with m edges admits a balanced bipartition V1, V2 of V(G) such that max{e(V1),e(V2)}

We prove that for any integers p≥k≥3 and any k-tuple of positive integers (n1,…,nk) such that p=∑i=1kni and n1≥n2≥⋯≥nk, the condition n1≤p2 is necessary and sufficient for every subgraph of the complete k-partite graph K(n1,…,nk) with at least 4-2p+2n1+∑i=1kni(p-ni)2edges to be chorded pancyclic. Removing all but one edge incident with any vertex of minimum degree in K(n1,…,nk) shows that this result is best possible. Our result implies that for any integers, k≥3 and n≥1, a balanced k-partite graph of order kn with has at least (k2-k)n2-2n(k-1)+42 edges is chorded pancyclic. In the case k=3, this result strengthens a previous one by Adamus, who in 2009 showed that a balanced tripartite graph of order 3n, n≥2, with at least 3n2-2n+2 edges is pancyclic. [ABSTRACT FROM AUTHOR]

Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed k≥1, among all graphs on n vertices with e edges, some threshold graph has the fewest matchings of size k; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call almost alternating threshold graphs. We also discuss a problem with a similar flavor: which threshold graph has the fewest independent sets. Here we are inspired by the result that among all graphs on n vertices and e edges the lex graph has the most independent sets. [ABSTRACT FROM AUTHOR]

Let G be an infinite family of connected graphs and let k be a positive integer. We say that k is forcing for G if for all G∈G but finitely many, the following holds. Any {-1,1}-weighing of the edges of G for which all connected subgraphs on k edges are positively weighted implies that G is positively weighted. Otherwise, we say that it is weakly forcing for G if any such weighing implies that the weight of G is bounded from below by a constant. Otherwise we say that kcollapses for G. We classify k for some of the most prominent classes of graphs, such as all connected graphs, all connected graphs with a given maximum degree and all connected graphs with a given average degree. [ABSTRACT FROM AUTHOR]

For graphs G and H and integer k≥1, the Gallai-Ramsey number grk(G:H) is defined to be the minimum integer N such that if KN is edge-colored with k colors, then there is either a rainbow G or a monochromatic H. It is known that grk(K3:C2n+1) is exponential in k. In this note, we improve the upper bound for grk(K3:C2n+1) obtained by Hall et al., and give bounds for grk(K3:Km,n). [ABSTRACT FROM AUTHOR]

Abstract The maximum matching width is a width-parameter that is defined on a branch-decomposition over the vertex set of a graph. The size of a maximum matching in the bipartite graph is used as a cut-function. In this paper, we characterize the graphs of maximum matching width at most 2 using the minor obstruction set. Also, we compute the exact value of the maximum matching width of a grid. [ABSTRACT FROM AUTHOR]

A map is said to be even-closed if all of its automorphisms act like even permutations on the vertex set. In this paper the study of even-closed regular maps is approached by analysing two distinguished families. The first family consists of embeddings of a well-known family of graphs on distinct orientable surfaces, whereas in the second family we consider all graphs having orientable-regular embeddings on a particular surface. In particular, the classification of even-closed orientable-regular embeddings of the complete bipartite graphs K n , n and classification of even-closed orientable-regular maps on the torus are given. [ABSTRACT FROM AUTHOR]

Many famous graphs are edge-primitive. Weiss (1973) and Guo et al. (2013) determined edge-primitive graphs of valency 3 and 5, respectively. In this paper, we study edge-primitive graphs of any prime valency. It is proved that all such graphs are 2-arc-transitive and the full automorphism groups are almost simple with the only exception the graphs being the complete bipartite graphs, and a complete classification is given of such graphs with soluble edge stabilizers, which widely extends the classifications of Weiss and Guo et al. (notice that the edge stabilizers of edge-primitive graphs with valency at most 5 are soluble). [ABSTRACT FROM AUTHOR]

Let G be a bipartite graph with adjacency matrix A. If G has a unique perfect matching, then A has an inverse A1 which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39]. [ABSTRACT FROM AUTHOR]

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon-Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area. [ABSTRACT FROM AUTHOR]

A vertex v of a graph G = (V, E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domination number γve(G). A ve-dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total ve-dominating set of G is the total vertex-edge domination number γ t ve(G). In this paper we initiate the study of total vertex-edge domination. We show that determining the number γ t ve(G) for bipartite graphs is NP-complete. Then we show that if T is a tree different from a star with order n, ℓ leaves and s support vertices, then γ t ve(T) ≤ (n - ℓ + s)/2. Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that γ t ve(G) = 2γve(G) and we provide a characterization of all trees T with γ t ve(T) = 2γve(T). [ABSTRACT FROM AUTHOR]

A forced cycle C of a graph G is a cycle in G such that G − V ( C ) has a unique perfect matching. A graph G is a cycle-forced graph if every cycle in G is a forced cycle. In this paper, we give a characterization of cycle-forced hamiltonian bipartite graphs. [ABSTRACT FROM AUTHOR]

We examine the quantity S ( G ) = ∑ u v ∈ E ( G ) min ( deg u , deg v ) over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of S ( G ) is achieved by three different classes of constructions, depending on the distance between the number of edges and the nearest triangular number. Furthermore we determine the maximum possible value when the set of graphs is restricted to be bipartite, a forest, or to be planar given sufficiently many edges. The quantity S ( G ) corresponds to the difference between two well studied indices, the irregularity of a graph and the sum of the squares of the degrees in a graph. These are known as the first and third Zagreb indices in the area of mathematical chemistry. [ABSTRACT FROM AUTHOR]

In recent years, the vertex connectivity and edge connectivity have aroused great interest of many scholars in the field of mathematics. We consider in this note that edge connectivity of a graph called k ′(G). In this note, If G is a bipartite graph with vertices n ≥ 3 we prove that k ′(G × Kn) = min {n(n - 1) k ′(G), (n - 1) δ(G)}, and their edge connectivity is satisfied k ′(G) = δ(G), we can obtain k ′(G × Kn) = δ (G × Kn) = (n - 1) δ(G). [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer, and let G be a graph. A function f : V ( G ) → 2 { 1 , … , k } is a k -rainbow dominating function of G if every vertex x ∈ V ( G ) with f ( x ) = ∅ satisfies ⋃ y ∈ N G ( x ) f ( y ) = { 1 , … , k } . The k -rainbow domination number of G , denoted by γ r k ( G ) , is the minimum weight w ( f ) = ∑ x ∈ V ( G ) | f ( x ) | of a k -rainbow dominating function f of G . In this paper, we prove that for every connected graph G of order n ≥ 8 with δ ( G ) ≥ 2 , γ r 3 ( G ) ≤ 5 n 6 . [ABSTRACT FROM AUTHOR]

Graph-token Petri net (GTPN), an extension of Tree-token Petri net (TTPN), is introduced by labelling the tokens with graphs. A study is done on the languages generated by GTPN. Some subclasses of graphs are generated by this Petri net model. [ABSTRACT FROM AUTHOR]

A -labeling (or - valuation) of a graph is an injection from the vertices of the graph with ‘qʹ edges to the set {0, 1, 2, 3, ..., (2q - 1), 2q}, where if the edge labels induced by the absolute value of the difference of the vertex labels are a1; a2; a3; ..., aq - 1, aq then ai = i or ai = (2q + 1 - i). . A shell graph, C(n; n - 3) is defined as a cycle Cn with (n - 3) chords sharing a common endpoint called the apex. In other words, a shell graph is the join of complete graph K1 and Pm, the path with m vertices. A subdivided shell graph is obtained from the shell graph G = Pm ∨ K1 by subdividing the edges in the path Pm of the shell graph. A subdivided shell ower graph is defined as a one vertex union of k copies of the subdivided shell graph and k copies of the complete graph K2. In this paper, we prove that subdivided shell ower graphs admit subdivided shell ower graph is defined as a one vertex union of k copies of the subdivided shell graph and k copies of the complete graph K2. In this paper, we prove that subdivided shell ower graphs admit ρ -labeling. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the extended duplicate graphs of path graph, comb graph and twig graph admit skolem difference mean labeling, skolem odd difference mean labeling and skolem even difference mean labeling. [ABSTRACT FROM AUTHOR]

A graceful labeling of a graph G with q edges is an injection f : V (G) → {0, 1, 2, ...,g} with the property that the resulting edge labels are also distinct, where an edge incident with the vertices u and v is assigned the label |f(u) - f (v)|. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove that the path union of vertex switching of even cycles in increasing order is graceful. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. A broadcast on G is a function f : V ( G ) → N such that f ( v ) ≤ e G ( v ) holds for every vertex v of G , where e G ( v ) denotes the eccentricity of v in G , that is, the maximum distance from v to any other vertex of G . The cost of f is the value cost ( f ) = ∑ v ∈ V ( G ) f ( v ) . A broadcast f on G is independent if for every two distinct vertices u and v in G , d G ( u , v ) > max { f ( u ) , f ( v ) } , where d G ( u , v ) denotes the distance between u and v in G . The broadcast independence number of G is then defined as the maximum cost of an independent broadcast on G . In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the broadcast independence number of caterpillars having no pair of adjacent trunks, a trunk being an internal spine vertex with degree 2. [ABSTRACT FROM AUTHOR]

A set S of vertices in a graph G is a total dominating set of G if every vertex has a neighbor in S . The total domination number, γ t ( G ) , is the minimum cardinality of a total dominating set of G . Let n G and m G denote the number of vertices and edges, respectively, in G . We prove that all the essential upper bounds on the total domination number of a graph G without isolated vertices and isolated edges can be written in the unified form γ t ( G ) ≤ ( 2 a n G + 2 b m G ) ∕ ( 3 a + 2 b ) for constants b ≥ 0 and a ≥ 2 3 ( 1 − b ) . We also provide unified forms of upper bounds for the total domination number of a graph with minimum degree δ in the cases when δ ∈ { 2 , 3 , 4 } . For example, we show that the bound γ t ( G ) ≤ a n G + b m G is valid for every graph G with minimum degree δ = 3 if and only if both b ≥ 0 and a ≥ 1 2 ( 1 − 3 b ) hold. [ABSTRACT FROM AUTHOR]