Pedro García-Vázquez

On the order of ({r,m};g)-cages of even girth

By an ({r,m};g)-cage we mean a graph on a minimum number of vertices f({r,m};g) with degree set {r,m}, 2==8. Moreover, we obtain for any integer k>=2 that f({r,k(r-1)+1};6)=2k(r-1)^2+2r where r-1 a is prime power. This result supports the conjecture that f({r,m};6)=2(rm-m+1) for any r

Connectivity of graphs with given girth pair

Girth pairs were introduced by Harary and Kovacs [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g,h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g,h) such that g is odd and h>=g+3 is even has high (vertex-)connectivity if its diameter is at most h-3. The edge version of all results is also studied.

On the super-restricted arc-connectivity of s -geodetic digraphs

For a strongly connected digraph D the restricted arc-connectivity λ′(D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D - S has a non-trivial strong component D1 such that D - V (D1) contains an arc. In this paper we prove that every digraph on at least 4 vertices and of minimum degree at least 2 is λ′ -connected and λ′(D) ≤ξ′(D), where ξ′(D) is the minimum arc-degree of D. Also in this paper we introduce the concept of super- λ′ digraphs and provide a sufficient condition for an s -geodetic digraph to be super- λ′. Further, we show that the h -iterated line digraph Lh(D) of an s -geodetic digraph is super- λ′ for a particular h.

Edge-connectivity in Pk-path graphs

Abstract The P k ( G ) -path graph corresponding to a graph G has for vertices the set of all paths of length k in G. Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k - 1 in G, and their union forms either a cycle or a path of length k + 1 . Path graphs were introduced by Broersma and Hoede (J. Graph. Theory 13 (1989) 427) as a generalization of line graphs, because for k = 1 , path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (Math. Ann. 182 (1969) 170), later by Zamfirescu (Math. Ann. 187 (1970) 305), and by Jixiang Meng (Graph Theory Notes of New York XL (2001) 12). The connectivity of P k -path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the P k ( G ) -path graphs to be disconnected, assuming that G has girth of at least k + 1 . Going one step further, we prove in this work that the edge-connectivity of P k ( G ) is at least λ ( P k ( G ) ) ⩾ δ ( G ) - 1 for a graph G of girth at least k + 1 and minimum degree δ ( G ) ⩾ 2 . Furthermore, we show λ ( P k ( G ) ) ⩾ 2 δ ( G ) - 2 provided that δ ( G ) ⩾ 3 .

Restricted arc-connectivity of generalized p-cycles

Abstract For a strongly connected digraph D the restricted arc-connectivity λ ′ ( D ) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D − S has a non-trivial strong component D 1 such that D − V ( D 1 ) contains an arc. In this paper we prove that a generalized p -cycle is λ ′ -optimal if diam ( D ) ≤ 2 l + p − 2 , where l is the semigirth of D and p ≥ 3 . Further we show that the k -iterated line digraph is λ ′ -optimal if diam ( D ) ≤ 2 l + p − 2 + k for p ≥ 3 . We improve these results for p large enough and we also improve known results on super- λ for p -cycles with p ≥ 3 .

Trees having an even or quasi even degree sequence are graceful

Abstract A rooted tree with diameter D is said to have an even degree sequence if every vertex has even degree except for one root and the leaves, which are in the last level ⌊ D / 2 ⌋ . The degree sequence is said to be quasi even if every vertex has even degree except for one root, every vertex in level ⌊ D / 2 ⌋ − 1 and the leaves, which are in the last level ⌊ D / 2 ⌋ . Hrnciar and Haviar [P. Hrnciar, A. Haviar, All trees of diameter five are graceful, Discrete Math. 233 (2001) 133–150] give a method to construct a graceful labeling for every tree with diameter five. Based upon their method we prove that every tree having an even or quasi even degree sequence is graceful. To do that we find for a tree of even diameter and rooted in its central vertex t of degree δ ( t ) up to δ ( t ) ! graceful labelings if the tree has an even or quasi even degree sequence.

On the connectivity and restricted edge-connectivity of 3-arc graphs

Let G ? denote the symmetric digraph of a graph G . A 3-arc is a 4-tuple ( y , a , b , x ) of vertices such that both ( y , a , b ) and ( a , b , x ) are paths of length 2 in G . The 3-arc graph X ( G ) of a given graph G is defined to have vertices the arcs of G ? , and they are denoted as ( u v ) . Two vertices ( a y ) , ( b x ) are adjacent in X ( G ) if and only if ( y , a , b , x ) is a 3-arc of G . The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs. We prove that the 3-arc graph X ( G ) of every connected graph G of minimum degree ? ( G ) ? 3 has λ ( X ( G ) ) ? ( ? ( G ) - 1 ) 2 . Furthermore, if G is a 2-connected graph, then X ( G ) has restricted edge-connectivity λ ( 2 ) ( X ( G ) ) ? 2 ( ? ( G ) - 1 ) 2 - 2 . We also provide examples showing that all these bounds are sharp. Concerning the vertex-connectivity, we prove that ? ( X ( G ) ) ? min { ? ( G ) ( ? ( G ) - 1 ) , ( ? ( G ) - 1 ) 2 } . This result improves a previous one by M. Knor, S. Zhou, Diameter and connectivity of 3-arc graphs, Discrete Math. 310 (2010) 37-42]. Finally, we obtain that X ( G ) is superconnected if G is maximally connected.

Toughness of the corona of two graphs

The toughness of a non-complete graph G=(V, E) is defined as τ(G)=min{|S|/ω(G−S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G−S) denotes the number of components of the resultant graph G−S by deletion of S. The corona of two graphs G and H, written as G° H, is the graph obtained by taking one copy of G and |V(G)| copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. In this paper, we investigate the toughness of this kind of graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, stars, wheels or complete graphs.

On extremal bipartite graphs with high girth

Abstract Let us denote by E X ( m , n ; { C 4 , … , C 2 t } ) the family of bipartite graphs G with m and n vertices in its classes that contain no cycles of length less than or equal to 2t and have maximum size. In this paper the following question is proposed: does always such an extremal graph G contain a ( 2 t + 2 ) -cycle? The answer is shown to be affirmative for t = 2 , 3 or whenever m and n are large enough in comparison with t. The latter asymptotical result needs two preliminary theorems. First we state that the diameter of an extremal bipartite graph is at most 2t, and afterwards we show that its girth is equal to 2 t + 2 when the minimum degree is at least 2 and the maximum degree is at least t + 1 . We also give the exact value of the extremal function e x ( m , n ; { C 4 , … , C 2 t } ) for m = n = 2 t and m = n = 2 t + 1 and show that all the extremal bipartite graphs of E X ( m , n ; { C 4 , … , C 2 t } ) are maximally connected.

Superconnectivity of graphs with odd girth g and even girth h

A maximally connected graph of minimum degree @d is said to be superconnected (for short super-@k) if all disconnecting sets of cardinality @d are the neighborhood of some vertex of degree @d. Sufficient conditions on the diameter to guarantee that a graph of odd girth g and even girth h>=g+3 is super-@k are stated. Also polarity graphs are shown to be super-@k.