Narong Punnim

Group Divisible Designs with Two Associate Classes and 2=1

The original classiffcation of PBIBDs defined group divisible designs GDD( 𝑣 = 𝑣 1 + 𝑣 2 + ⋯ + 𝑣 𝑔 , 𝑔 , 𝑘 , 𝜆 1 , 𝜆 2 ) with 𝜆 1 ≠ 0 . In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with 𝜆 2 = 1 .

The Hamiltonian Number of Cubic Graphs

A Hamiltonian walk in a connected graph G of order n is a closed spanning walk of minimum length in G . The Hamiltonian number h (G ) of a connected graph G is the length of a Hamiltonian walk in G . Thus h may be considered as a measure of how far a given graph is from being Hamiltonian. We prove that if G runs over the set of connected cubic graphs of order n and

The decycling number of cubic graphs

n\not= 14

Switchings, realizations, and interpolation theorems for graph parameters

then the values h (G ) completely cover a line segment [a ,b ] of positive integers. For an even integer n *** 4, let

The decycling number of cubic planar graphs

\mathcal{C}(3^n)

Degree Sequences and Chromatic Numbers of Graphs

be the set of all connected cubic graphs of order n . We define

The Clique Numbers of Regular Graphs

\min(h, 3^n)=\min\{h(G):G\in \mathcal{C}(3^n)\}

On Non 3-Choosable Bipartite Graphs

and

GDDs with Two Associate Classes and with Three Groups of Sizes 1, n, n and λ1<λ2

\max(h, 3^n)=\max\{h(G):G\in \mathcal{C}(3^n)\}

An Intermediate Value Theorem for the Arboricities

. Thus for an even integer n *** 4, the two invariants min (h , 3 n ) and max (h , 3 n ) naturally arise. Evidently, min (h , 3 n ) = n . The exact values of max (h , 3 n ) are obtained in all situations.