E. Abajo

New small regular graphs of girth 5

A (k,g)-graph is a k-regular graph with girth g and a (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k,g)-graphs of minimum order n(k,g). We focus on girth g=5, where cages are known only for degrees k7. We construct (k,5)-graphs using techniques exposed by Funk (2009) and Abreu et al. (2012) to obtain the best upper bounds on n(k,5) known hitherto. The tables given in the introduction show the improvements obtained with our results.

The Menger number of the strong product of graphs

The x y -Menger number with respect to a given integer ? , for every two vertices x , y in a connected graph G , denoted by ? ? ( x , y ) , is the maximum number of internally disjoint x y -paths whose lengths are at most ? in G . The Menger number of G with respect to ? is defined as ? ? ( G ) = min { ? ? ( x , y ) : x , y ? V ( G ) } . In this paper we focus on the Menger number of the strong product G 1 ? G 2 of two connected graphs G 1 and G 2 with at least three vertices. We show that ? ? ( G 1 ? G 2 ) ? ? ? ( G 1 ) ? ? ( G 2 ) and furthermore, that ? ? + 2 ( G 1 ? G 2 ) ? ? ? ( G 1 ) ? ? ( G 2 ) + ? ? ( G 1 ) + ? ? ( G 2 ) if both G 1 and G 2 have girth at least 5. These bounds are best possible, and in particular, we prove that the last inequality is reached when G 1 and G 2 are maximally connected graphs.

Size of Graphs with High Girth

Abstract Let n ⩾ 4 be a positive integer and let e x ( ν ; { C 3 , … , C n } ) denote the maximum number of edges in a { C 3 , … , C n } -free simple graph of order ν. This paper gives the exact value of this function for all ν up to ⌊ ( 16 n − 15 ) / 5 ⌋ . This result allows us to deduce all the different values of the girths that such extremal graphs can have. Let k ⩾ 0 be an integer. For each n ⩾ 2 log 2 ( k + 2 ) there exists ν such that every extremal graph G with e ( G ) − ν ( G ) = k has minimal degree at most 2, and is obtained by adding vertices of degree 1 and/or by subdividing a graph or a multigraph H with δ ( H ) ⩾ 3 and e ( H ) − ν ( H ) = k .

Elliptic semiplanes and regular graphs with girth 5

Abstract A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k, g)-graphs of minimum order n(k, g). We focus on girth g = 5 , where cages are known only for degrees k ≤ 7 . Considering the relationship between finite geometries and graphs we establish upper constructive bounds on n(k, 5), for k ∈ { 13 , 14 , 17 , 18 , … } that improve the best so far known.