Juan Carlos Valenzuela

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

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 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.

Extremal K(s,t)-free bipartite graphs

Abstract We approach a well-known topic in extremal graph theory, the so-called Zarankiewicz Problem [K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951), 301], which consists in: (i) calculating the maximum number of edges z ( m , n ; s , t ) that a bipartite graph G with partite classes of cardinalities m and n can have such that G is free of a complete bipartite subgraph K ( s , t ) with s vertices in the m class and t vertices in the n class; (ii) describing all the corresponding extremal bipartite graphs having that maximum number of edges. In this paper, the exact value of z ( m , n ; s , t ) is calculated and the corresponding family Z ( m , n ; s , t ) of extremal graphs is characterized when the parameters satisfy certain relationships.

Highly connected star product graphs

Abstract In this work we approach the connectivity κ of a kind of product graphs that were introduced by J.C. Bermond et al. in 1984. More precisely, we provide lower bounds for κ, and state sufficient conditions that guarantee these product graphs to be maximally connected or superconnected. A main consequence is that even graphs with low connectivity may lead to highly connected larger (product) graphs.

On the order of bi-regular cages of even girth

Abstract By a bi-regular cage of girth g we mean a graph with prescribed degrees r and m and with the least possible number of vertices denoted by f ( { r , m } ; g ) . We provide new upper and lower bounds of f ( { r , m } ; g ) for even girth g ⩾ 6 . Moreover, we prove that f ( { r , k ( r − 1 ) + 1 } ; 6 ) = 2 k ( r − 1 ) 2 + 2 r where k ⩾ 2 is any integer and r − 1 is a prime power. This result supports the conjecture f ( { r , m } ; 6 ) = 2 ( r m − m + 1 ) for any r m formulated by Yuansheng and Liang (The minimum number of vertices with girth 6 and degree set D = { r , m } , Discrete Mathematics 269 (2003), 249–258).