Graphs G1 Research Articles

Simple cubic graphs with no short traveling salesman tour

Let tsp(G) denote the length of a shortest traveling salesman tour in a graph G. We prove that for any ε>0, there exists a simple 2-connected planar cubic graph G1 such that tsp(G1)≥(1.25−ε)⋅|V(G1)|, a simple 2-connected bipartite cubic graph G2 such that tsp(G2)≥(1.2−ε)⋅|V(G2)|, and a simple 3-connected cubic graph G3 such that tsp(G3)≥(1.125−ε)⋅|V(G3)|.

Packing Graphs of Bounded Codegree

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumption that G1 or G2 has bounded codegree. In particular, we prove for all t ⩾ 2 that if G1 contains no copy of the complete bipartite graph K2,t and Δ(G1) > 17t · Δ(G2), then (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1 implies that G1 and G2 pack. We also provide a mild improvement if moreover G2 contains no copy of the complete tripartite graph K1,1,s, s ⩾ 1.

A Sharp Dirac–Erdős Type Bound for Large Graphs

Let k ⩾ 3 be an integer, hk(G) be the number of vertices of degree at least 2k in a graph G, and ℓk(G) be the number of vertices of degree at most 2k − 2 in G. Dirac and Erdős proved in 1963 that if hk(G) − ℓk(G) ⩾ k2 + 2k − 4, then G contains k vertex-disjoint cycles. For each k ⩾ 2, they also showed an infinite sequence of graphs Gk(n) with hk(Gk(n)) − ℓk(Gk(n)) = 2k − 1 such that Gk(n) does not have k disjoint cycles. Recently, the authors proved that, for k ⩾ 2, a bound of 3k is sufficient to guarantee the existence of k disjoint cycles, and presented for every k a graph G0(k) with hk(G0(k)) − ℓk(G0(k)) = 3k − 1 and no k disjoint cycles. The goal of this paper is to refine and sharpen this result. We show that the Dirac–Erdős construction is optimal in the sense that for every k ⩾ 2, there are only finitely many graphs G with hk(G) − ℓk(G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and hk(G) − ℓk(G) ⩾ 2k contains k disjoint cycles.

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs G1,G2,…,Gk, the bipartite Ramsey number b(G1,G2,…,Gk) is the least positive integer b so that any coloring of the edges of Kb,b with k colors will result in a copy of Gi in the ith color for some i. In this paper, our main focus will be to bound the following numbers: b(C2t1,C2t2) and b(C2t1,C2t2,C2t3) for all ti≥3,b(C2t1,C2t2,C2t3,C2t4) for 3≤ti≤9, and b(C2t1,C2t2,C2t3,C2t4,C2t5) for 3≤ti≤5. Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.

The Very Cost Effective Graph Folding of the Join of Two Graphs

In this paper, we studied the very cost effective graph property for the join graph of two graphs. In general this is may or may not be a very cost effective graph. We obtained the conditions for the join graph of two graphs to be a very cost effective graph. First we proved that the join graph Pn∨Pm of path graphs is very cost effective graph if n+m is an even number and is not if n+m is an odd number. Then we proved that the join graph of any two cycle graphs Cn and Cm where n, m are both odd is very cost effective, and the join graph Pn∨Cn is a very cost effective graph if n is an odd number. Also we proved that the join graph G1∨G2 of two very cost effective graphs G1 and G2 is a very cost effective graph if n(G1) + n(G2) is even. Finally we proved that the graph folding of the join graph of two very cost effective graphs not always very cost effective but this will be the case if the sum of the numbers of the vertices in the image of the graph folding is even.

The harmonic index of edge-semitotal graphs, total graphs and related sums

For a connected graph G, there are several related graphs such as line graph L(G), subdivision graph S(G), vertex-semitotal graph R(G), edge-semitotal graph Q(G) and total graph T(G) [I. Gutman, B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5-16, W. Yan, B.-Y. Yang and Y. -N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett. 20 (2007), 290-295]. Let F be one of symbols S, R, Q or T. The F-sum G1+FG2 of two connected graphs G1 and G2 is a graph with vertex set (V (G1) U E(G1)) × V (G2) in which two vertices (u1, v1) and (u2, v2) of G1 +F G2 are adjacent if and only if [u1 = u2 _ V (G1) and u1v2 _ E(G2)] or [v1 = v2 and u1u2 2 E(F(G))] [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009), 794-803]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and F-sum of graphs, where F = Q or T.

List colouring of graphs and generalized Dyck paths

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume G is a graph and f:V(G)→N is a mapping. For a nonnegative integer m, let f(m) be the extension of f to the graph G▪ Km¯ for which f(m)(v)=|V(G)| for each vertex v of Km¯. Let mc(G,f) be the minimum m such that G▪ Km¯ is not f(m)-choosable and mp(G,f) be the minimum m such that G▪ Km¯ is not f(m)-paintable. We study the parameter mc(Kn,f) and mp(Kn,f) for arbitrary mappings f. For x→=(x1,x2,…,xn), an x→-dominated path ending at (a,b) is a monotonic path P of the a×b grid from (0,0) to (a,b) such that each vertex (i,j) on P satisfies i≤xj+1. Let ψ(x→) be the number of x→-dominated paths ending at (xn,n). By this definition, the Catalan number Cn equals ψ((0,1,…,n−1)). This paper proves that if G=Kn has vertices v1,v2,…,vn and f(v1)≤f(v2)≤…≤f(vn), then mc(G,f)=mp(G,f)=ψ(x→(f)), where x→(f)=(x1,x2,…,xn) and xi=f(vi)−i for i=1,2,…,n. Therefore, if f(vi)=n, then mc(Kn,f)=mp(Kn,f) equals the Catalan number Cn. We also show that if G=G1∪G2∪⋯∪Gp is the disjoint union of graphs G1,G2,…,Gp and f=f1∪f2∪⋯∪fp, then mc(G,f)=∏i=1pmc(Gi,fi) and mp(G,f)=∏i=1pmp(Gi,fi). This generalizes a result in Carraher et al. (2014), where the case each Gi is a copy of K1 is considered.

Maximizing the spectral radius of graphs with fixed minimum degree and edge connectivity

The spectral radius ρ(G) of a graph G is the largest eigenvalue of the adjacency matrix A(G). Suppose a graph G0 maximizes the spectral radius over the class of graphs of order n with fixed minimum degree δ and edge connectivity κ′<δ. In this paper, we mainly show that G0≅Bn,δκ′, where Bn,δκ′ is obtained by adding κ′ edges between Kδ+1 and Kn−δ−1. A property of the adjacency matrix of G0 is also obtained. Moreover, graphs that maximize ρ(G) over the class of graphs with minimum degree δ and edge-connectivity κ′, for κ′=0,1,2,3,δ, are completely determined.

The adjacency spectrum of two new operations of graphs

Let G be a graph and A=A(G) be its adjacency matrix. The eigenvalues μ1,μ2,…,μn of A(G) are the eigenvalues of G and form the adjacency spectrum, denoted by Spec(G). In this paper, we introduce two new operations G1■k(G3□G2) and (G4□G1)■k(G3□G2), and describe the adjacency spectra of G1■k(G3□G2) and (G4□G1)■k(G3□G2) of regular graphs G1, G2 and arbitrarily graphs G3, G4 in terms of their adjacency spectra. As the applications, we obtain some new integral spectrum graphs.

A note on hajos stable graphs

Let G1 and G2 be two undirected graphs, ( v w ) be an edge of G1, and ( x y ) be an edge of G2. Hajos construction forms a new graph H, that combines the two graphs by identifying vertices v and x into a single vertex, removing the two edges ( v w ) and ( x y ), and adding a new edge ( w y ). In this paper we have discussed the properties of graphs G1 and G2 that are Hajos stable and not Hajos stable graphs. We have proved that, if G1 and G2 are Euler graphs, then the Hajos graph is also Euler. We have obtained conditions whether the Hajos stable graphs are Hamiltonian and not Hamiltonian graphs. Also we have provided on NG type result for Hajos stable graphs. We have proved that, if G1 and G2 are just excellent and Hajos stable then H is not just excellent. Also we have shown that very excellent graphs are not Hajos stable graphs.

The common-neighbourhood of a graph

The vulnerability measures on a connected graph which are mostly used and known are based on the Neighbourhood concept. Neighbour-integrity, edge-integrity and accessibility number are some of these measures. In this work we define and examine the Common-neighbourhood of a connected graph as a new global connectivity measure. It takes account the neighbourhoods of all pairs of vertices. We show that, for connected graphs G1 and G2 of order n, if the dominating number of G1 is bigger than the dominating number of G2, then the common- neighbourhood of G1 is less than the common-neighbourhood of G2. We prove some theorems on common-neighbourhood of a graph.

Stars versus stripes Ramsey numbers

For given simple graphs G1,G2,…,Gt, the Ramsey number R(G1,G2,…,Gt) is the smallest positive integer n such that if the edges of the complete graph Kn are partitioned into t disjoint color classes giving t graphs H1,H2,…,Ht, then at least one Hi has a subgraph isomorphic to Gi. In this paper, for positive integers t1,t2,…,ts and n1,n2,…,nc the Ramsey number R(St1,St2,…,Sts,n1K2,n2K2,…,ncK2) is computed exactly, where nK2 denotes a matching (stripe) of size n, i.e., n pairwise disjoint edges and Sn is a star with n edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gyárfás and Sárközy.

Locating a robber with multiple probes

We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager (2012). Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph G there is a winning strategy for the cop on the graph G1∕m obtained by replacing each edge of G by a path of length m, if m≥n (Carragher et al., 2012). The present authors showed that, for all but a few small values of n, this bound may be improved to m≥n∕2, which is best possible (Haslegrave et al., 2016). In this paper we consider the natural extension in which the cop probes a set of k vertices, rather than a single vertex, at each turn. We consider the relationship between the value of k required to ensure victory on the original graph with the length of subdivisions required to ensure victory with k=1. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of k for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree Δ, which is best possible up to a factor of (1−o(1)).

Packing two graphs of even girth 10

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that, if (Δ(G1)+1)(Δ(G2)+1)≤n+1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumptions that neither G1 nor G2 contain a 4-, 6- or 8-cycle, and that Δ(G1) or Δ(G2) is large enough (≥940060).

On the complexity of various parameterizations of common induced subgraph isomorphism

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G1 and G2, one looks for a graph with the maximum number of vertices being both an induced subgraph of G1 and G2. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of Clique, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by k:=vc(G1)+vc(G2) the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time 2O(klog⁡k). We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2o(klog⁡k). This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by k, unless NP⊆coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.

Subnormality of unbounded composition operators over one-circuit directed graphs: Exotic examples

A recent example of a non-hyponormal injective composition operator in an L2-space generating Stieltjes moment sequences, invented by three of the present authors, was built over a non-locally finite directed tree. The main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and, in the affirmative case, to find the simplest possible graph with these properties (simplicity refers to local valency). The problem is solved affirmatively for the locally finite directed graph G2,0, which consists of two branches and one loop. The only simpler directed graph for which the problem remains unsolved consists of one branch and one loop. The consistency condition, the only efficient tool for verifying subnormality of unbounded composition operators, is intensively studied in the context of G2,0, which leads to a constructive method of solving the problem. The method itself is partly based on transforming the Krein and the Friedrichs measures coming either from shifted Al-Salam–Carlitz q-polynomials or from a quartic birth and death process.

Near packings of two graphs

A packing of two graph G1 and G2 is a set {H1,H2} such that H1≅G1, H2≅G2, and H1 and H2 are edge disjoint subgraphs of Kn. Let F be a family of graphs. A F-near-packing of G1 and G2 is a generalization of a packing. In a F-near-packing, H1 and H2 may overlap so the subgraph defined by the edges common to both of them is a member of F. In the paper we study three families of graphs (1) Ek –the family of all graphs with at most k edges, (2) Dk –the family of all graphs with maximum degree at most k, and (3) Kk –the family of all graphs having the clique number at most k. By m(n,F) we denote the smallest number m such that there are graphs G1 and G2 both of order n with |E(G1)|+|E(G2)|=m which do not have a F-near-packing. It is well known that m(n,E0)=m(n,D0)=m(n,K1)=3(n−1)2+1 because a E0, D0 or K1-near-packing is just a packing. In this paper we prove that m(n,E1)=m(n,D1)=2n−1 and, for sufficiently large n, m(n,D2)=5(n−1)2+1. We also obtain distinct bounds on m(n,Ek) for k≥2, m(n,D3), and m(n,Kk) for k≥2, under conditions imposed on n and k.

Polarity graphs and Ramsey numbers for [formula omitted] versus stars

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) shows that R(C4,K1,n)≤n+⌊n−1⌋+2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R(C4,K1,q2−t)=q2+q−(t−1) if q is an odd prime power, 1≤t≤2⌈q4⌉ and t≠2⌈q4⌉−1, which extends a result on R(C4,K1,q2−t) obtained by Parsons (1976).

Unimodality of the independence polynomials of some composite graphs

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ &gt; 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

The Energy and Operations of Graphs

Let G be a finite and undirected simple graph on n vertices, A(G) is the adjacency matrix of G, λ1,λ2,...,λn are eigenvalues of A(G), then the energy of G is . In this paper, we determine the energy of graphs obtained from a graph by other unary operations, or graphs obtained from two graphs by other binary operations. In terms of binary operation, we prove that the energy of product graphs is equal to the product of the energy of graphs G1 and G2, and give the computational formulas of the energy of Corona graph , join graph of two regular graphs G and H, respectively. In terms of unary operation, we give the computational formulas of the energy of the duplication graph DmG, the line graph L(G), the subdivision graph S(G), and the total graph T(G) of a regular graph G, respectively. In particularly, we obtained a lot of graphs pair of equienergetic.