Journal of Combinatorial Mathematics and Combinatorial Computing

ISSN: 0835-3026 (print) 2817-576X (online)

The Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC) embarked on its publishing journey in April 1987. From 2024 onward, it publishes four volumes per year in March, June, September and December. JCMCC has gained recognition and visibility in the academic community and is indexed in renowned databases such as MathSciNet, Zentralblatt, Engineering Village and Scopus. The scope of the journal includes; Combinatorial Mathematics, Combinatorial Computing, Artificial Intelligence and applications of Artificial Intelligence in various files.

Ahmad M. Assaf1
1Department of Mathematics Central Michigan University Mt. Pleasant, MI U.S.A. 48859
Abstract:

Let \(V\) be a finite set of order \(v\). A \((v, k, \lambda)\) covering design of index \(\lambda\) and block size \(k\) is a collection of \(k\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks, \(\alpha(v, k, \lambda)\), in a covering design. It is well known that \(\alpha(v,k,\lambda) \geq \lceil\frac{v}{k}\lceil\frac{v-1}{k-1}\lambda\rceil\rceil = \phi(v, k, \lambda)\), where \(\lceil x \rceil\) is the smallest integer satisfying \(x \leq \lceil x \rceil\). It is shown here that \(\alpha(v,5,7) = \phi(v, 5, 7)\) for all positive integers \(v \geq 5\) with the possible exception of \(v = 22, 28, 142, 162\).

Xiang-Ying Su1
1Department of Mathematics Wayne State University Detroit, MI 48202, USA
Abstract:

A graph \(G\) is called \(k\)-critical if \(\chi(G) = k\) and \(\chi(G – e) k\) is at most \(n – k + 3\) if \(k \leq 7\).

Yan Guiying1
1Department of Mathematics Shandong University, Jinan Shandong 250100 P.R. of China
Abstract:

Let \(g\) and \(f\) be integer-valued functions defined on \(V(G)\) with \(f(v) \geq g(v) \geq 1\) for all \(v \in V(G)\). A graph \(G\) is called a \((g, f)\)-graph if \(g(v) \leq d_G(v) \leq f(v)\) for each vertex \(v \in V(G)\), and a \((g, f)\)-factor of a graph \(G\) is a spanning \((g, f)\)-subgraph of \(G\). A graph is \((g, f)\)-factorable if its edges can be decomposed into \((g, f)\)-factors.
The purpose of this paper is to prove the following three theorems: (i) If \(m \geq 2\), every \(\left((2mg+2m-2)t+(g+1)s, (2mf-2m+2)t+(f-1)s\right)\)-graph \(G\) is \((g, f)\)-factorable. (ii) Let \(g(x)\) be even and \(m > 2\). (1) If \(m\) is even, and \(G\) is a \(\left((2mg+2)t+(g+1)s, (2mf-2m+4)t+(f-1)s\right)\)-graph, then \(G\) is \((g, f)\)-factorable; (2) If \(m\) is odd, and \(G\) is a \(((2mg+4)t+(g+1)s$, $(2mf-2m+2)t+(f-1)s)\)-graph, then \(G\) is \((g, f)\)-factorable. (iii) Let \(f(x)\) be even and \(m > 2\). (1) If \(m\) is even, and \(G\) is a \(\left((2mg+2m-4)t+(g+1)s, (2mf-2)t+(f-1)s\right)\)-graph, then \(G\) is \((g, f)\)-factorable;
(2) If \(m\) is odd, and \(G\) is a \(((2mg+2m-2)t+(g+1)s\), \((2mf-4)t+(f-1)s)\)-graph, then \(G\) is \((g, f)\)-factorable.
where \(t\), \(m\) are integers and \(s\) is a nonnegative integer.

Dragomir Z. Dokovic1
1Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1
Abstract:

All Williamson matrices in this Note are symmetric circulants. Eight non-equivalent sets of Williamson matrices of order \(25\) are known. They were discovered by Williamson (\(2\) sets), Baumert and Hall (\(2\) sets), and Sawade (\(4\) sets). Sawade carried out a complete search and reported that there are exactly eight non-equivalent such sets of matrices. Subsequently, this was confirmed by Koukouvinos and Kounias. It is surprising that we have found two more such sets. Hence, there are ten non-equivalent sets of Williamson matrices of order \(25\).

Only three non-equivalent sets of Williamson matrices of order \(37\) were known so far. One of them was discovered by each of Williamson, Turyn, and Yamada. We have found one more such set.

R. G. Stanton1, D. W. Kroeker1
1Department of Computer Science University of Manitoba Winnipeg, Canada R3T 2N2
D. V. Chopra1, R. Dios2
1Wichita State University Wichita, Kansas
2New Jersey Institute of Technology Newark, New Jersey
Abstract:

In this paper, we derive and present some necessary conditions for the existence of certain combinatorial arrays (called balanced arrays (\(B\)-arrays)) with two elements by making use of some classical inequalities. We discuss briefly the usefulness of these arrays in combinatorics and statistical design of experiments.

E. J. Farrell1, M.A. Sam Chee1
1 Department of Mathematics The University of the West Indies St Augustine, Trinidad
Abstract:

An explicit recurrence is obtained for the clique polynomial of a short ladder in which the two diagonals are drawn in each cell. From this result, an explicit formula for the number of decompositions of the ladder into triangles and \(4\)-cliques is obtained. The recurrence is then used to obtain results for the matching polynomial of the ladder. Finally, an association is made with a particular tiling problem.

Dalibor Froncek1
1Department of Mathematics and Statistics McMaster University Hamilton, Ontario Canada L8S 4K1
Abstract:

Let \(G\) be a finite graph and \(x\) be its vertex. The \({neighbourhood}\) of \(x\) in \(G\), denoted \(N_G(x)\), is a subgraph of \(G\) induced by all vertices adjacent to \(x\). \(G\) is a \({graph \; with \; a \; constant \; neighbourhood}\) if there exists a graph \(H\) such that \(N_G(x)\) is isomorphic to \(H\) for every vertex \(x\) of \(G\).

We completely characterize graphs with constant neighbourhoods isomorphic to complements of regular disconnected graphs.

M. E. Bascufién 1, S. Ruiz1, R. C. Brigham 2, R. M. Caron2, P. J. Slater3, RP. Vitray4
1Universidad Catélica de Valparaiso, Chile
2Department of Mathematics and Computer Science University of Central Florida Orlando, Florida 32816
3Department of Mathematics University of Alabama in Huntsville Huntsville, Alabama 35899
4Department of Mathematics Rollins College Winter Park, Florida 32789
Abstract:

A \({numbering}\) of a graph \(G = (V, E)\) is a bijection \(f: V \rightarrow \{1, 2, \ldots, p\}\) where \(|V| = p\). The \({additive \; bandwidth \; of \; numbering}\) \(f\) is \(B^+(G, f) = \max\{|f(u) + f(v) – (p + 1)| : uv \in E\}\), and the \({additive \; bandwidth}\) of \(G\) is \(B^+(G) = \min\{B^+(G, f) : f \text{ a numbering of } G\}\). Labeling \(V\) by a numbering which yields \(B^+(G)\) has the effect of causing the \(1\)’s in the adjacency matrix of \(G\) to be placed as near as possible to the main counterdiagonal, a fact which offers potential storage savings for some classes of graphs. Properties of additive bandwidth are discussed, including relationships with other graphical invariants, its value for cycles, and bounds on its value for extensions of full \(k\)-ary trees.

Puhua Guan1, Tayuan Huang2
1Department of Mathematics University of Puerto Rico Rio Piedras. PR 00931
2Department of Applied Mathematics National Chiao-Tung University Hsin-Chu 30050, Taiwan, ROC
Abstract:

Let \(\Gamma_\ell\) be the union of \(n\) complete graphs \(A_1, A_2, \ldots, A_n\) of size \(n\) each such that \(|A_i \cap A_j| \leq \ell\) whenever \(i \neq j\). We prove that the chromatic number of \(\Gamma_\ell\) is bounded above by \((2n – 4)\ell + 1\).

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