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Topological indices play a significant role in crisp, fuzzy graphs and their real-life application. But to avoid the vagueness in the final result, these indices should be dealt with in the neutrosophic environment since it consolidates the uncertain quantity or values of an event in the name of "indeterminacy membership". Except for the wiener index, no other indices are introduced under the neutrosophic graphical setting. In this article, we consider the forgotten topological index (ToI) and the Edge forgotten index in the 3-valued logic neutrosophic graph and came up with some important theorem results and applications. [ABSTRACT FROM AUTHOR]

We examine the classic game of Cops and Robbers played on dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the (static) underlying graph. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n2k+1T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs with at most a linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n -- 1+ λ edges. Furthermore, we define a family of underlying graphs with n -- 1 + λ edges where λ -- 1 cops are necessary (and sufficient) for capture. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the YSU A: Physical & Mathematical Sciences is the property of Publishing House of Yerevan State University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Morris and Saxton used the method of containers to bound the number of n‐vertex graphs with m edges containing no ℓ‐cycles, and hence graphs of girth more than ℓ. We consider a generalization to r‐uniform hypergraphs. The girth of a hypergraph H is the minimum ℓ≥2 such that there exist distinct vertices v1,...,vℓ and hyperedges e1,...,eℓ with vi,vi+1∈ei for all 1≤i≤ℓ. Letting Nmr(n,ℓ) denote the number of n‐vertex r‐uniform hypergraphs with m edges and girth larger than ℓ and defining λ=⌈(r−2)∕(ℓ−2)⌉, we show Nmr(n,ℓ)≤Nm2(n,ℓ)r−1+λ, which is tight when ℓ−2 divides r−2 up to a 1+o(1) term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than ℓ in random r‐uniform hypergraphs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new parameter of domination in graphs called ”Rings Domination Number”. Some properties and important boundaries of the rings dominating set and rings domination number have been discussed. Also, this number for certain graphs have been determined. Furthermore, some operations on two graphs as join, composition, cross product and corona, have been introduced and determined the rings domination numbers for each one of them. [ABSTRACT FROM AUTHOR]

For any finite simple graph G one can associate a simplicial complex Δ (G) and an edge ideal I(G). We attempt to use the combinatorial properties of the given graph to understand some of the algebraic properties of the associated edge ideal. In particular, we study some important algebraic properties like graded Betti numbers, Castelnuovo–Mumford regularity and projective dimension of the edge ideal associated to the given ladder graph. We will introduce the new concept of A-type subgraphs and edge-disjoint A-type subgraphs to reduce the complexity of computing the regularity of a ladder graph. We are also able to find the projective dimension of a ladder graph using the Nguyen–Vu and Kimura's works on the projective dimension of edge ideals. [ABSTRACT FROM AUTHOR]

In the ▪- Free Edge Deletion problem (CDFED), we are given a graph G and an integer k > 0 , and the question is whether there are at most k edges whose deletion results in a graph without claws and diamonds as induced subgraphs. Based on some refined observations, we propose a kernel of O (k 3) vertices and O (k 4) edges, significantly improving the previous kernel of O (k 12) vertices and O (k 24) edges. In addition, we derive an O ⁎ (3.792 k) -time algorithm for CDFED. [ABSTRACT FROM AUTHOR]

We consider homological edge percolation on a sequence ( G t ) t of finite graphs covered by an infinite (quasi)transitive graph H and weakly convergent to H. In particular, we use the covering maps to classify 1-cycles on graphs G t as homologically trivial or non-trivial and define several thresholds associated with the rank of thus defined first homology group on the open subgraphs generated by the Bernoulli (edge) percolation process. We identify the growth of the homological distance dt, the smallest size of a non-trivial cycle on G t , as the main factor determining the location of homology-changing thresholds. In particular, we show that the giant cycle erasure threshold p E 0 (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold p c (H) if the ratio dt/ln nt diverges, where nt is the number of edges of G t , and we give evidence that p E 0 < p c (H) in several cases where this ratio remains bounded, which is necessarily the case if H is non-amenable. [ABSTRACT FROM AUTHOR]

A subset S of vertices in a graph G = (V, E) is 2-independent if every vertex of S has at most one neighbor in S. The 2-independence number is the maximum cardinality of a 2-independent set of G. In this paper, we initiate the study of the 2-independence subdivision number sdβ2 (G) defined as the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-independence number. We first show that for every connected graph G of order at least three, 1 ≤ β2 (G) ≤ 2, and we give a necessary and sufficient condition for graphs G attaining each bound. Moreover, restricted to the class of trees, we provide a constructive characterization of all trees T with sdβ2(T) = 2, and we show that such a characterization suggests an algorithm that determines whether a tree T has β2 (T) = 2 or sd^2 (T) = 1 in polynomial time. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer, and let G be a graph. A P≥k-factor of a graph G is a spanning subgraph F of G such that each component of F is a path of order at least k. A graph G is a P≥k-factor uniform graph if G has a P≥k-factor including e1 and excluding e2 for any two distinct edges e1 and e2 of G. In this article, we verify that a 3-edge-connected graph G is a P≥3-factor uniform graph if its sun toughness s(G) > 1. Furthermore, we show that the two conditions on edge-connectivity and sun toughness are sharp. [ABSTRACT FROM AUTHOR]

• For more than 37.5 percent of g ≤ ⌊ 3 n 2 ⌋ , the g -extra edge-connectivity of ternary n -cube, λ g (Q n 3) , exists a concentration behavior. • Determine the exact value of λ g (Q n 3) for each 3 ⌈ n 2 ⌉ + r − ⌊ 3 2 r + e + 1 2 ⌋ ≤ g ≤ 3 ⌈ n 2 ⌉ + r , r = 0 , 1 , ... , ⌊ n 2 ⌋ − 1. • For the above each integer g , a necessary and sufficient condition of λ g -optimality is found. • Obtain maximum number of edge-disjoint paths connecting any two disjoint connected subgraphs with the above g vertices in Q n 3. One of most central issues in various interconnection networks of parallel and distributed systems is to find edge-disjoint paths concerned with an information transmission through links. It is universally acknowledged that an interconnection network can be modeled as an undirected simple graph G = (V , E) with processors and physical links between the processors represented as vertices and edges of the G , respectively. The studies on the edge-disjoint paths are closely related to the edge-connectivity. By the well-known Menger theorem, the maximum number of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g -extra edge-connectivity of G λ g (G). It is the cardinality of the minimum set of edges in G , if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The k -ary n -cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. This article intends to show that for 3 ⌈ n 2 ⌉ + r − ⌊ 3 2 r + e + 1 2 ⌋ ≤ g ≤ 3 ⌈ n 2 ⌉ + r , the g -extra edge-connectivity of ternary n -cube (also called 3-ary n -cube) (n ≥ 3) exists a concentration behavior, and prove that these corresponding g -extra edge-connectivities of ternary n -cubes are the constants 2 (⌊ n 2 ⌋ − r) 3 ⌈ n 2 ⌉ + r , where r = 0 , 1 , 2 , ⋯ , ⌊ n 2 ⌋ − 1 , and e = 1 if n is odd and e = 0 if n is even. The above upper and lower bounds of g are sharp. As the integer g varies within the range between 3 ⌈ n 2 ⌉ + r − ⌊ 3 2 r + e + 1 2 ⌋ and 3 ⌈ n 2 ⌉ + r , a necessary and sufficient condition of λ g -optimality is found. Our results improve the several previous results. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and m edges. The minimum edge dominating energy is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph G. In this paper, we establish some bounds for the minimum edge dominating energy of subgraphs of a graph. [ABSTRACT FROM AUTHOR]

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph T consisting of three edges e, f and g such that | e ∩ f | = | f ∩ g | = | g ∩ e | = 1 and e ∩ f ∩ g = ∅ . We prove that if H is an n-vertex r-uniform hypergraph with maximum degree ▵ , then as ▵ → ∞ , the number of edges in a densest T-free subhypergraph of H is at least e (H) ▵ r - 2 r - 1 + o (1) . For r = 3 , this is tight up to the o (1) term in the exponent. We also show that if H is a random n-vertex triple system with edge-probability p such that p n 3 → ∞ as n → ∞ , then with high probability as n → ∞ , the number of edges in a densest T-free subhypergraph is min (1 - o (1)) p n 3 , p 1 3 n 2 - o (1) . We use the method of containers together with probabilistic methods and a connection to the extremal problem for arithmetic progressions of length three due to Ruzsa and Szemerédi. [ABSTRACT FROM AUTHOR]

Let G be a graph that may have multiedges, let μ (u , v) denote the number of edges joining u and v in G, and let μ (u) = max v ∈ N (u) μ (u , v) . Let a , b : V (G) ⟶ N \ { 0 , 1 } be two functions with d G (v) ≥ a (v) + b (v) + 2 μ (v) - 3 for each v ∈ V (G) . We show that, if G does not contain C 4 , then G admits a partition (A, B) with d G [ A ] (u) ≥ a (u) for each u ∈ A , and d G [ B ] (v) ≥ b (v) for each v ∈ B . This generalizes a theorem of Ma and Yang on simple graphs, and answers a question of Schweser and Stiebitz. [ABSTRACT FROM AUTHOR]

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph H , there exists a function f such that for every 4-edge-connected graph G , either G contains k pairwise edge-disjoint subgraphs each containing H as an immersion, or there exists a set of at most f (k) edges of G intersecting all such subgraphs. This theorem is best possible in the sense that the 4-edge-connectivity cannot be replaced by the 3-edge-connectivity. [ABSTRACT FROM AUTHOR]

For a graph G an edge-covering of G is a family of subgraphs H1,H2, . . ., Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi, i = 1, 2, . . ., t. In this case we say that G admits an (H1,H2, . . ., Ht)-(edge) covering. An H-covering of graph G is an (H1,H2, . . ., Ht)-(edge) covering in which every subgraph Hi is isomorphic to a given graph H. Let G be a graph admitting H-covering. An edge k-labeling: E(G) = {1, 2, . . ., k} is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H' and H" isomorphic to H their weights wt(H') and wt(H") are distinct. The weight of a subgraph H under an edge k-labeling is the sum of labels of edges belonging to H. The edge H-irregularity strength of a graph G, denoted by ehs(G,H), is the smallest integer k such that G has an H-irregular edge k-labeling. In this paper we determine the exact values of ehs(G,H) for prisms, antiprisms, triangular ladders, diagonal ladders, wheels and gear graphs. Moreover the subgraph H is isomorphic to only C4, C3 and K4. [ABSTRACT FROM AUTHOR]

We obtain a new bound on the number of edges in induced subgraphs of Johnson graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a graph. A set M ⊆ E(G) is a matching if no two edges in M share a common vertex. A matching of G is perfect if it covers every vertex in G. A matching of a graph G with odd order is called a near perfect matching if it has edges. In this paper, we completely characterize the forbidden subgraph and pairs of forbidden subgraphs that force a 2-connected graph with a (near) perfect matching to be hamiltonian. [ABSTRACT FROM AUTHOR]

The edge clique cover number ecc(G) of a graph G is the size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw‐free, then ecc(G) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in n for ecc(G) for such graphs, improving upon the bou nd of O(n4∕3log1∕3 n) due to Javadi, Maleki, and Omoomi. More precisely we prove that ecc(G) is at most the minimum of n+δ(G) and 2n−Ω(nlog n), where δ(G) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 32n. We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ecc(G)) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4. [ABSTRACT FROM AUTHOR]

Among the classical models for interconnection networks are hypercubes and Fibonacci cubes. Fibonacci cubes are induced subgraphs of hypercubes obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s, counted by Fibonacci numbers. Another set of binary strings which are counted by Fibonacci numbers are those with a restriction on the runlengths. Induced subgraphs of the hypercube on the latter strings as vertices define Fibonacci-run graphs. They have the same number of vertices as Fibonacci cubes, but fewer edges and different graph theoretical properties. We obtain properties of Fibonacci-run graphs including the number of edges, the analogue of the fundamental recursion, the average degree of a vertex, Hamiltonicity, special degree sequences, and the number of hypercubes they contain. A detailed study of the degree sequences of Fibonacci-run graphs is interesting in its own right and is reported in a companion paper. [ABSTRACT FROM AUTHOR]

A k-bisection of a graph is a partition of the vertices in two classes whose cardinalities differ of at most one and such that the subgraphs induced by each class are acyclic with all connected components of order at most k. Esperet, Tarsi and the second author proved in 2017 that every simple cubic graph admits a 3-bisection. Recently, Cui and Liu extended that result to the class of simple subcubic graphs. Their proof is an adaptation of the quite long proof of the cubic case to the subcubic one. Here, we propose an easier proof of a slightly stronger result. Indeed, starting from the result for simple cubic graphs, we prove the existence of a 3-bisection for all cubic graphs (also admitting parallel edges). Then we prove the same result for the larger class of subcubic graphs as an easy corollary. [ABSTRACT FROM AUTHOR]

Let G be a complete convex geometric graph, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that has an edge in common with every element of F . In Keller and Perles (Discrete Comput Geom 60(1):1–8, 2018) we gave an explicit description of all blockers for the family of simple (i.e., non-crossing) Hamiltonian paths (SHPs) in G in the 'even' case | V (G) | = 2 m . It turned out that all the blockers are simple caterpillar trees of a certain class. In this paper we give an explicit description of all blockers for the family of SHPs in the 'odd' case | V (G) | = 2 m - 1 . In this case, the structure of the blockers is more complex, and in particular, they are not necessarily simple. Correspondingly, the proof is more complicated. [ABSTRACT FROM AUTHOR]

The goal of an outdegree-constrained edge-modification problem is to find a spanning subgraph or supergraph H of an input undirected graph G such that either: (Type I) the number of edges in H is minimized or maximized and H can be oriented to satisfy some specified constraints on the vertices' resulting outdegrees; or: (Type II) among all subgraphs or supergraphs of G that can be constructed by deleting or inserting a fixed number of edges, H admits an orientation optimizing some objective involving the vertices' outdegrees. This paper introduces eight new outdegree-constrained edge-modification problems related to load balancing called (Type I) MIN-DEL-MAX, MIN-INS-MIN, MAX-INS-MAX, and MAX-DEL-MIN and (Type II) p -DEL-MAX, p -INS-MIN, p -INS-MAX, and p -DEL-MIN. In each of the eight problems, the input is a graph and the goal is to delete or insert edges so that the resulting graph has an orientation in which the maximum outdegree (taken over all vertices) is small or the minimum outdegree is large. We first present a framework that provides algorithms for solving all eight problems in polynomial time on unweighted graphs. Next we investigate the inapproximability of the edge-weighted versions of the problems, and design polynomial-time algorithms for six of the problems on edge-weighted trees. [ABSTRACT FROM AUTHOR]

For k ≥ 1 and n ≥ 2 k , the Kneser graph K G (n , k) has all k -element subsets of an n -element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of K G (n , k) is n − 2 k + 2. Schrijver constructed a vertex-critical subgraph S G (n , k) of K G (n , k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for k = 2 and arbitrary n ≥ 4 by means of a simple explicit combinatorial definition. [ABSTRACT FROM AUTHOR]

A graph is 1‐planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1‐planar. For a 3‐connected locally maximal 1‐planar graph G, we show the existence of a spanning 3‐connected planar subgraph and prove that G is Hamiltonian if G has at most three 3‐vertex‐cuts, and that G is traceable if G has at most four 3‐vertex‐cuts. Moreover, infinitely many nontraceable 5‐connected 1‐planar graphs are presented. [ABSTRACT FROM AUTHOR]

We investigate 4-chromatic Schrijver graphs from various points of view and color-critical edges in Schrijver graphs in general. In particular, we present the following results. We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus providing such a proof also for 4-chromatic Kneser graphs. We show that only certain types of edges of Schrijver graphs can be color-critical and prove that many of those are indeed color-critical, though in general not all of them are. In the 4-chromatic case these results give a complete characterization of the color-critical edges. We show that a spanning subgraph of 4-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special case of a result by Kaiser and Stehlík. We show that (apart from two cases of small parameters) the subgraphs we present that quadrangulate the Klein bottle are edge-color-critical. The analogous result for the subgraphs quadrangulating the projective plane is an immediate consequence of earlier results by Gimbel and Thomassen and was already noted by Kaiser and Stehlík. Our proof of non-3-colorability of 4-chromatic Schrijver graphs is based on a complete description of their structure that we present as well. This was already also given (in somewhat different terms) by Braun and even earlier in an unpublished manuscript by Li. [ABSTRACT FROM AUTHOR]

Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider k -Fibonacci cubes, which we obtain as subgraphs of Fibonacci cubes by eliminating certain edges during the fundamental recursion phase of their construction. These graphs have the same number of vertices as Fibonacci cubes, but their edge sets are determined by a parameter k. We obtain properties of k -Fibonacci cubes including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain. [ABSTRACT FROM AUTHOR]

We give the crossing number of the join product $W_{4}+D_{n}$ , where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once. [ABSTRACT FROM AUTHOR]

A class of graphs is χ-bounded if there is a function f such that χ (G) ≤ f (ω (G)) for every induced subgraph G of every graph in the class, where χ , ω denote the chromatic number and clique number of G respectively. In 1987, Gyárfás conjectured that for every c , if C is a class of graphs such that χ (G) ≤ ω (G) + c for every induced subgraph G of every graph in the class, then the class of complements of members of C is χ -bounded. We prove this conjecture. Indeed, more generally, a class of graphs is χ -bounded if it has the property that no graph in the class has c + 1 odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if C is a shortest odd hole in a graph, and X is the set of vertices with at least five neighbours in V (C) , then there is a three-vertex set that dominates X. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin Institute of Technology. Social Sciences Edition / Haerbin Gongye Daxue Xuebao. Shehui Kexue Ban is the property of Harbin Institute of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k ⋅ P 3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k ⋅ P 3 as a subgraph, respectively. [ABSTRACT FROM AUTHOR]

The classical problem of estimating the number of edges in a subgraph of a special distance graph is considered. Old results are significantly improved. [ABSTRACT FROM AUTHOR]

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained. [ABSTRACT FROM AUTHOR]

The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every d -regular graph, d ≥ 2 , can be decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1–2–3 Conjecture if d ∉ { 10 , 11 , 12 , 13 , 15 , 17 } , and into at most 3 such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most 24 subgraphs fulfilling the 1–2–3 Conjecture, improving the previously best upper bound of 40. Both results are partly based on applications of the Lovász Local Lemma. [ABSTRACT FROM AUTHOR]

Let Γ be a triangulation of a connected closed 2-dimensional (not necessarily orientable) surface. Using zigzags (closed left–right walks), for every face of Γ we define the z -monodromy which acts on the oriented edges of this face. There are precisely 7 types of z -monodromies. We consider the following two cases: (M1) the z -monodromy is identity, (M2) the z -monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph Γ ∗ formed by edges whose z -monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of z -knotted triangulations. [ABSTRACT FROM AUTHOR]

As spanning subgraphs of hypercubes, exchanged hypercubes contain less edges but maintain lots of desired properties of hypercubes. This paper considers matching preclusion, a kind of measures of edge-fault tolerance, of exchanged hypercubes EH(s, t). We show that EH(s, t) is maximally matched, that is, for s ≥ t, mp(EH(s, t)) = t + 1 and EH(s, t) is super matched if and only if (s, t) ≠ (1, 1). Comparing with results of matching preclusion for hypercubes, we conclude that exchanged hypercubes maintain the matching preclusion property of hypercubes, except for EH(1, 1). [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in the polynomial ring In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete k-partite graph, fan graph, multi-fan graph and wheel graph. [ABSTRACT FROM AUTHOR]

For a graph G, let f 2 (G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f 2 (G) over 3-regular n-vertex simple graphs G. To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max { 0 , ⌊ (c - 1) / 2 ⌋ } vertices. More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max { 0 , ⌊ (3 n - 2 m + c - 1) / 2 ⌋ } vertices. These bounds are sharp; we describe the extremal multigraphs. [ABSTRACT FROM AUTHOR]

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger's result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connected graph, in Catlin et al. (2009), it is shown that every 4-edge-connected graph remains collapsible after removing any two edges. We prove the following. • [(i)] Every 4-edge-connected G contains two vertices x , y such that one of x and y has minimum degree in G and both G − x and G − y are collapsible. • [(ii)] Let G be a 4-edge-connected graph and let X ⊂ E (G) be an edge subset with | X | ≤ 3. Then G − X is collapsible if and only if X is not contained in a 4-edge-cut of G. • [(iii)] Let G be a 4-edge-connected graph and let X ⊂ E (G) be an edge subset with | X | ≤ 4. Then G − X is collapsible if and only if G − X is not contractible to a member in { K 2 c , K 2 , K 2 , 2 , K 2 , 3 , K 2 , 4 }. These extend former results of Jaeger (1979) and Catlin (1988). [ABSTRACT FROM AUTHOR]

Abstract If k ≥ 0 , then a k -edge-coloring of a graph G is an assignment of colors to edges of G from the set of k colors, so that adjacent edges receive different colors. A k -edge-colorable subgraph of G is maximum if it is the largest among all k -edge-colorable subgraphs of G. For a graph G and k ≥ 0 , let ν k (G) be the number of edges of a maximum k -edge-colorable subgraph of G. In 2010 Mkrtchyan et al. proved that if G is a cubic graph, then ν 2 (G) ≤ | V | + 2 ν 3 (G) 4. This result implies that if the cubic graph G contains a perfect matching, in particular when it is bridgeless, then ν 2 (G) ≤ ν 1 (G) + ν 3 (G) 2. One may wonder whether there are other interesting graph-classes, where a relation between ν 2 (G) and ν 1 (G) + ν 3 (G) 2 can be proved. Related with this question, in this paper we show that ν k (G) ≥ ν k − i (G) + ν k + i (G) 2 for any bipartite graph G , k ≥ 0 and i = 0 , 1 , ... , k. [ABSTRACT FROM AUTHOR]

We obtain new estimates for the number of edges in induced subgraphs of some distance graph. [ABSTRACT FROM AUTHOR]

Let G be an edge‐coloured graph. The minimum colour degree δc(G) of G is the largest integer k such that, for every vertex v, there are at least k distinct colours on edges incident to v. We say that G is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any ε>0 and n large, every edge‐coloured graph G with δc(G)≥(1∕2+ε)n contains a properly coloured cycle of length at least min{n,⌊2δc(G)∕3⌋}. [ABSTRACT FROM AUTHOR]

Abstract For a set of graphs F , the extremal number e x (n ; F) is the maximum number of edges in a graph of order n not containing any subgraph isomorphic to some graph in F. If F contains a graph on n vertices, then we often call the problem a spanning Turán problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let L n k be the set of all linear forests of order n with at least n − k + 1 edges. We prove that when n ≥ 3 k and k ≥ 2 , e x (n ; L n k) = n − k + 1 2 + O (k 2). Clearly, the result is interesting when k = o (n). [ABSTRACT FROM AUTHOR]

A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,...,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,...,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,...,Gm, and then joining every vertex u in Gi, i∈{1,2,...,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles. [ABSTRACT FROM AUTHOR]

We show how to define a plethora of metrics for graphs—either full graphs or subgraphs. The method mainly utilizes the minimal matching between any two multisets of positive real numbers by comparing the multiple edges with respect to their corresponding vertices. In the end of this article, we also demonstrate how to implement these defined metrics with the help of adjacency matrices. These metrics are easy to be manipulated in real applications and could be amended according to different situations. By our metrics, one should be able to compare the distances between graphs, trees, and networks, in particular those with fuzzy properties. [ABSTRACT FROM AUTHOR]

Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex‐deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes. [ABSTRACT FROM AUTHOR]

Abstract A weighted graph is called stable if the maximum weight of an integral matching equals the cost of a minimum-weight fractional vertex cover. We address the following question: how can we modify a given unstable graph in the least intrusive manner in order to achieve stability? Previous works have addressed stabilization through addition or deletion of the smallest possible number of edges/vertices. In this work we investigate the following more fine-grained additive stabilization strategy: given a graph G = (V , E) with unit edge weights; find non-negative c ∈ R E with minimum ∑ e c e such that adding c e to the unit edge weight of each e ∈ E yields a stable graph. We provide the first super-constant hardness of approximation results for any graph stabilization problem: (i) unless the current best-known algorithm for the densest- k -subgraph problem can be improved, there is no o (| V | 1 ∕ 24) -approximation for additive stabilizers; (ii) the additive stabilizer problem has no o (log | V |) approximation unless P = N P. On the algorithmic side, we present (iii) a polynomial time algorithm with approximation factor at most | V | for a super-class of the instances generated in our hardness proofs, (iv) an algorithm to solve min additive stabilizer in factor-critical graphs exactly in polynomial time, and (v) an algorithm to solve min additive stabilizer in arbitrary graphs exactly in time exponential in the size of the Tutte set. Our main tools are the Gallai–Edmonds decomposition and structural results for the problem that reduce the continuous decision domain to a discrete decision domain. [ABSTRACT FROM AUTHOR]

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from the complement of G to G results in a copy of H. The minimum number of edges (the size) of an H-saturated graph on n vertices is denoted sat(n,H), while the maximum size is the well studied extremal number, ex(n,H). The saturation spectrum for a graph H is the set of sizes of H-saturated graphs between sat(n,H) and ex(n,H). In this paper we show that paths, trees with a vertex adjacent to many leaves, and brooms have a gap in the saturation spectrum. [ABSTRACT FROM AUTHOR]

Abstract Let Γ be a graph with n vertices, where each edge is given an orientation and let Q be the vertex–edge incidence matrix of Γ. Suppose that Γ has a cut-vertex v and Γ − v = Γ [ V 1 ] ∪ Γ [ V 2 ]. We obtain a relation between the Moore–Penrose inverse of the incidence matrix of Γ and of the incidence matrices of the induced subgraphs Γ [ V 1 ∪ { v } ] and Γ [ V 2 ∪ { v } ]. The result is used to give a combinatorial interpretation of the Moore–Penrose inverse of the incidence matrix of a graph whose blocks are either cliques or cycles. Moreover we obtain a description of minors of the Moore–Penrose inverse of the incidence matrix when the rows are indexed by cut-edges. The results generalize corresponding results for trees in the literature. [ABSTRACT FROM AUTHOR]