SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER

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1 Discussiones Mathematicae Gaph Theoy 39 (019) doi: /dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann Lehstuhl II fü Mathematik RWTH Aachen Univesity 5056 Aachen, Gemany Abstact Let G be a connected gaph with minimum degee δ and edge-connectivity λ. A gaph is maximally edge-connected if λ = δ, and it is supe-edgeconnected if evey minimum edge-cut is tivial; that is, if evey minimum edge-cut consists of edges incident with a vetex of minimum degee. The clique numbe ω(g) of a gaph G is the maximum cadinality of a complete subgaph of G. In this pape, we show that a connected gaph G with clique numbe ω(g) is maximally edge-connected o supe-edge-connected if the numbe of edges is lage enough. These ae genealizations of coesponding esults fo tiangle-fee gaphs by Volkmann and Hong in 017. Keywods: edge-connectivity, clique numbe, maximally edge-connected gaphs, supe-edge-connected gaphs. 010 Mathematics Subject Classification: 05C Teminology and Intoduction Let G be a finite and simple gaph with vetex set V = V(G) and edge set E = E(G). The ode and size of G ae defined by n = n(g) = V(G) and m = m(g) = E(G), espectively. If N(v) = N G (v) is the neighbohood of the vetex v V(G), then we denote by d(v) = d G (v) = N(v) the degee of v and by δ = δ(g) the minimum degee of the gaph G. Fo a subset X V(G), let G[X] to denote the subgaph of G induced by X. Fo two subsets X and Y of V(G) let [X,Y] be the set of edges with one endpoint in X and the othe one in Y. The clique numbe ω(g) of a gaph G is the maximum cadinality of a

2 568 L. Volkmann complete subgaph of G. An edge-cut of a connected gaph G is a set of edges whose emoval disconnects G. The edge connectivity λ = λ(g) of a connected gaph G is defined as the minimum cadinality of an edge-cut ove all edge-cuts of G. An edge-cut S is a minimum edge-cut o a λ-cut if S = λ(g). The inequality λ(g) δ(g) is immediate. We call a connected gaph maximally edge-connected, if λ(g) = δ(g). In 1981, Baue et al. [1] poposed the concept of supe-edge connectedness. A gaph is called supe-edge-connected o supe-λ if evey minimum edge-cut is tivial; that is, if evey minimum edge-cut consists of edges incident with a vetex of minimum degee. Thus evey supe-edgeconnected gaph is also maximally edge-connected. Sufficient conditions fo gaphs to be maximally edge-connected o supeedge-connected wee given by seveal authos, see fo example the suvey pape by Hellwig and Volkmann [3]. The stating point was an aticle by Chatand [] in He obseved that if δ is lage enough, then the gaph is maximally edge-connected. A simila condition fo supe-edge-connectivity was given by Kelmans [4] six yeas late. Ove the yeas, these esults have been stengthened many times and in many ways. Recently, Volkmann and Hong [6] showed that a connected gaph o a connected tiangle-fee gaph is maximally edge-connected o supe-λ if the numbe of edges is lage enough. In paticula, they eceived the following esults. Theoem 1. Let G be a connected tiangle-fee gaph of ode n, size m, minimum degee δ and edge-connectivity λ. If then λ = δ. n m > δ(n 1 δ) 1, 4 Theoem. Let G be a connected tiangle-fee gaph of ode n, size m, minimum degee δ 3 and edge-connectivity λ. If then G is supe-λ. (n+1) m > δ(n+1 δ), 4 In this pape, we will genealize Theoems 1 and to connected gaphs with clique numbe ω(g) fo. Examples will demonstate that ou esults ae shap.

3 Sufficient Conditions fo Maximally Edge-Connected Gaphs Maximally Edge-Connected Gaphs The main tool of ou aticle is the famous theoem of Tuán [5]. Theoem 3. Let 1 be an intege, and let G be a gaph of ode n. If the clique numbe ω(g), then 1 E(G) n. Theoem 4. Let be an intege, and let G be a connected gaph of ode n, size m, minimum degee δ 1, edge-connectivity λ and clique numbe ω(g). If ( 1 δ ) δ m > n + n +δ 1, 1 1 then λ = δ. Poof. If δ = 1, then λ = δ in evey case. Thus assume in the following that δ. Suppose to the contay that λ δ 1. Then thee exist two disjoint sets X,Y V(G) with X Y = V(G) and [X,Y] = λ. Assume, without loss of geneality, that X Y. We fist show that X contains at least δ + 1 vetices. Othewise, suppose that X contains at most δ vetices. Then we obtain δ X x Xd G (x) X ( X 1)+λ δ( X 1)+δ 1. Obviously, this is a contadiction and thus X δ + 1. Using Theoem 3, we conclude that ( 1) X (1) E(G[X]) and ( 1) Y () E(G[Y]). Next we show that X (δ)/( 1). Suppose to the contay that X (δ)/( 1) 1. Since E(G[X]) = x X d G(x) λ, (1) implies that X δ x Xd ( 1) X ( 1) X G (x) +λ +δ 1 X 1 ( ) δ 1 +δ 1 X 1 ( ) δ δ 1 = X δ 1 X +δ 1

4 570 L. Volkmann and thus X (δ 1) 1. Using this agument once moe, we aive at ( 1) X X δ +δ 1 X 1 = X (δ 1)+δ 1 (δ 1) 1 +δ 1 and thus X δ 1, which contadicts the fact that X δ + 1. Hence X (δ)/( 1). Since X + Y = n and X n/, the inequalities (1) and () lead to m= E(G[X]) + E(G[Y]) +λ ( 1) X ( 1) Y + +δ 1 ( 1) X ( 1)(n X ) = + +δ 1 ( 1) ( X +(n X ) ) +δ 1 ( 1) ( = n +( X n X ) ) +δ 1 ( ( 1) δ ) δ n + n +δ 1, 1 1 a contadiction to the hypothesis. Thus λ = δ. Theoem 1 is the special case = of Theoem 4. The next family of gaphs shows that Theoem 4 is best possible in the sense that ( 1 δ ) δ m = n + n +δ does not guaantee λ = δ. Example 5. Let and q 1 be integes. Let H 1 and H be two disjoint copies of the complete -patite gaph with q vetices in each patite set. Define H as the union of H 1 and H by adding δ 1 = q( 1) 1 edges between H 1 and H such that ω(h). Then H has ode n = q, minimum degee δ = q( 1) such that m(h)=q ( 1)+q( 1) 1 ( 1 δ ) δ = n + n +δ 1, 1 1 but obviously, λ(h) = δ(h) 1.

5 Sufficient Conditions fo Maximally Edge-Connected Gaphs Supe Edge-Connected Gaphs Theoem 6. Let be an intege, and let G be a connected gaph of ode n, size m, minimum degee δ, edge-connectivity λ and ω(g). If δ 3 o 3 and ( ( 1 δ ( δ m > n + 1) n 1) ) +δ, 1 1 then G is supe-λ. Poof. Suppose to the contay that G is not supe-λ. Then thee exist two disjoint sets X,Y V(G) such that X Y = V(G), X, Y and [X,Y] = λ. Assume, without loss of geneality, that X Y. We fist show that X contains at least δ vetices. Othewise, suppose that X contains at most δ 1 vetices. Then we obtain δ X x Xd G (x) X ( X 1 ) +λ (δ 1) ( X 1 ) +δ, which implies that X 1, contadicting that X. Thus X δ. Next we show that X (δ)/( 1) 1. If δ = and 3, then X δ = ()/( 1) 1 = (δ)/( 1) 1. Let now δ 3. Suppose to the contay that X contains at most (δ)/( 1) vetices. Since E(G[X]) = x X d G(x) λ, we conclude fom (1) that X δ x Xd ( 1) X ( 1) X G (x) +λ +δ X 1 ( ) δ +δ X 1 ( ) δ 1 1 +δ ( 1) = X δ X +δ and thus X δ ( 1). Using this agument once moe, we aive at X δ ( 1) X +δ X 1 δ X δ +δ = +δ ( 1) and thus X, which contadicts the fact that 3 δ X. Hence we have shown that X (δ)/( 1) 1 when δ 3 o 3. Since X + Y = n

6 57 L. Volkmann and X n/, the inequalities (1) and () lead to m = E(G[X]) + E(G[Y]) ( 1) X +λ ( 1) X ( 1)(n X ) = + +δ ( 1) ( X +(n X ) ) +δ ( 1) ( = n +( X n X ) ) +δ ( ( 1) n + + ( 1) Y ( δ ( δ 1) n 1) ) +δ, 1 1 a contadiction to the hypothesis. Thus G is supe-λ. +δ Theoem is the special case = of Theoem 6. The next family of gaphs shows that Theoem 6 is best possible in the sense that ( ( 1 δ ( δ m = n + 1) n 1) ) +δ 1 1 does not guaantee that the gaph is supe-λ. Example 7. Let and q 3 be integes. Let H 1 be the complete -patite gaph with q 1 vetices in one patite set and q vetices in 1 patite sets, and let H be the complete -patite gaph with q vetices in each patite set. Define H as the union of H 1 and H by adding δ = q( 1) edges between H 1 and H such that ω(h) and δ(h) = δ = q( 1). Then H has ode n = q 1, minimum degee δ = q( 1) such that m(h) = q q ( ( 1 δ ( δ = n + 1) n 1) ) +δ 1 1 but obviously, H is not supe-λ. Ou last example demonstates that Theoem 6 is not valid fo δ = and = in geneal. Example 8. Let q be an intege, and let K q,q be the complete bipatite gaph with the patite sets X = {x 1,x,...,x q } and Y = {y 1,y,...,y q }, and let u and v be two futhe vetices. Define the gaph H as the union of K q,q, u and

7 Sufficient Conditions fo Maximally Edge-Connected Gaphs v togethe with the edges uv, ux 1 and vx. Then H has ode n(h) = q +, minimum degee δ(h) = and ω(h). Futhemoe, m(h)=q +3 > q q +4 ( 1 = n + but H is not supe-λ. ( δ 1 ( δ 1) n 1) ) +δ 1 Refeences [1] D. Baue, C. Suffel, F. Boesch and R. Tindell, Connectivity extemal poblems and the design of eliable pobabilistic netwoks, in: The Theoy and Applications of Gaphs, Kalamazoo MI (Wiley, New Yok, 1981) [] G. Chatand, A gaph-theoetic appoach to a communications poblem, SIAM J. Appl. Math. 14 (1966) doi: / [3] A. Hellwig and L. Volkmann, Maximally edge-connected and vetex-connected gaphs and digaphs: A suvey, Discete Math. 308 (008) doi: /j.disc [4] A.K. Kelmans Asymptotic fomulas fo the pobability of k-connectedness of andom gaphs, Theoy Pobab. Appl. 17 (197) doi: / [5] P. Tuán, On an extemal poblem in gaph theoy, Mat. Fiz. Lapok 48 (1941) , in Hungaian. [6] L. Volkmann and Z.-M. Hong, Sufficient conditions fo maximally edge-connected and supe-edge-connected gaphs, Commun. Comb. Optim. (017) Received 8 June 017 Revised 19 Octobe 017 Accepted 3 Octobe 017

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi

Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated