A study of the total chromatic number of equibipartite graphs

AbstractThe total chromatic number χt(G) of a graph G is the least number of colors needed to color the vertices and edges of G so that no adjacent vertices or edges receive the same color, no incident edges receive the same color as either of the vertices it is incident with. In this paper, we obtain some results of the total chromatic number of the equibipartite graphs of order 2n with maximum degree n − 1. As a part of our results, we disprove the biconformability conjecture

A study of the total chromatic number of equibipartite graphs

AbstractThe total chromatic number χt(G) of a graph G is the least number of colors needed to color the vertices and edges of G so that no adjacent vertices or edges receive the same color, no incident edges receive the same color as either of the vertices it is incident with. In this paper, we obtain some results of the total chromatic number of the equibipartite graphs of order 2n with maximum degree n − 1. As a part of our results, we disprove the biconformability conjecture

Generalized Tur\'an problems for disjoint copies of graphs

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.Comment: 18 pages. There was a wrong statement in the first version, it is corrected no

Extremal Problems for Forests in Graphs and Hypergraphs

The Turan number, ex_r(n; F), of an r-uniform hypergraph F is the maximum number of hyperedges in an n-vertex r-uniform hypergraph which does not contain F as a subhypergraph. Note that when r = 2, ex_r(n; F) = ex(n; F) which is the Turan number of graph F. We study. Turan numbers in the degenerate case for graphs and hypergraphs; we focus on the case when F is a forest in graphs and hypergraph. In the first chapter we discuss the history of Turan numbers and give several classical results. In the second chapter, we examine the Turan number for forests with path components, forests of path and star components, and forests with all components of order 5. In the third chapter we determine the Turan number of an r-uniform star forest in various hypergraph settings

Generalized Turán problems for disjoint copies of a graph

Given two graphs H and F, the maximum possible number of copies of H in an F-free graph on n vertices is denoted by ex(n, H, F). We investigate the function ex(n, H, kF), where kF denotes k vertex disjoint copies of a fixed graph F. Our results include cases when F is a complete graph, cycle or a complete bipartite graph

Polychromatic colorings of certain subgraphs of complete graphs and maximum densities of substructures of a hypercube

If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. For H, K, subsets of the vertex set V(Qd) of the d-cube Qd, K is an exact copy of H if there is an automorphism of Qd sending H to K. For a positive integer, d, and a configuration in Qd, H, we define λ(H,d) as the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, λ(H,d) for several configurations in Q2, Q3, and Q4, and an infinite family of configurations. Strong connections exist with extensions Ramsey numbers for cycles in a graph, counting sequences with certain properties, inducibility of graphs, and we determine the inducibility of two vertex disjoint edges in the family of bipartite graphs