A bipartite graph G is called (α, β)-biregular if all vertices in one part of G have the degree α and all vertices in the other part have the degree β. An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored ...

We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. A (κ, λ) bipartite graph is a bipartite graph with left side of size κ and right side of size λ. We show that there is a homogeneous (א0, 2א0) bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending o...

We investigate some new results for strongly multiplicative labeling of graph. We prove that the graph obtained by arbitrary supersubdivision of tree T , grid graph Pn × Pm, complete bipartite graph Km,n, Cn Pm and one-point union of m cycle of length n are strongly multiplicative.

In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an n-vertex graph with O(n) edges and (n/ log n) maximal complete bipartite subgraphs Kk,`. ∗Work supported in part by NSF grant CCR-9258355.

The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree ∆ and diameterD. Bipartite graphs of maximum degree ∆, diameterD and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D = 2, 3, 4 and 6, and for D = 3, 4, 6, they have been constructed only for those values of ∆ such that ∆− 1 is ...

We present the Rainbow Code, a state-of-the-art genetic algorithm (GA) representation that encodes layered trees or bipartite trees (spanning trees of a pre-specified complete layered graph or complete bipartite graph) as integer strings of a certain form. The Rainbow Code possesses perfect feasibility, full coverage and zero bias, because the correspondence it defines between trees and strings...

Let G be a simple graph on the vertex set [n] with edge set E(G) and let S be the polynomial ring K[x1, . . . , xn, y1, . . . , yn] in 2n variables endowed with the lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn. The binomial edge ideal JG ⊂ S associated with G is generated by all the binomials fij = xiyj−xjyi with {i, j} ∈ E(G). The binomial edge ideals were introduced in [5]...

Let f(n, H) denote the maximum number of copies H possible in an n-vertex planar graph. The function has been determined when is a cycle length 3 or 4 by Hakimi and Schmeichel complete bipartite graph with smaller part size 1 2 Alon Caro. We determine exactly case path 3.

Let F be a family of graphs and let d be large enough. For every d-regular graph G, we study the existence of a spanning F -free subgraph of G with large minimum degree. This problem is wellunderstood if F does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we s...

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...