Hamiltonian cycles in particular k-partite graphs

Let $\text{G}$(itk, p) denote the class of k-partite graphs, where each part is a stable set of cardinality p and where the edges between any pair of stable sets are those of a perfect matching. Maru?i? has conjectured that if G belongs to $\text{G}$(k, p) and is connected then G is hamiltonian. It is proved that the conjecture is true for k ≤ 3 or p ≤ 3; but for k ≥ 4 and p ≥ 4 a non-hamiltonian connected graph in $\text{G}$(k, p) is constructed.     

Lower bounds on the independence number in terms of the degrees

Wei discovered that the independence number of a graph G is at least Σ_v(1 + d(v))^?1. It is proved here that if G is a connected triangle-free graph on n ≥ 3 vertices and if G is neither an odd cycle nor an odd path, then the bound above can be increased by $\text{n}\text{Δ(Δ + 1)}$, where Δ is the maximum degree. This new bound is sharp for even cycles and for three other graphs. These results relate nicely to some algorithms for finding large independent sets. They also have a natural matrix theory interpretation. A survey of other known lower bounds on the independence number is presented.     

Minimally k-connected graphs of low order and maximal size

Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) $\text{e(G)?kn?(}\text{k+1}\text{2}\text{)}$; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph K_k,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove $\text{e(G)??}\text{(n+k)}{}^2\text{8}\text{?}$ and characterize all minimally k-connected graphs of order n and size $\text{?(}\text{(n+k)}{}^2\text{8}\text{?}$.     

A note on the least prime in an arithmetic progression

Let k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(mod k). Let P(k) be the maximum value of p(k, l) for all l. We show $\text{lim inf}\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{≥ e}{}^{\mathrm{\gamma }}\text{= 1.78107…}$, where γ is Euler's constant and ? is Euler's function. We also show $\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{→ ∞}$ for almost all k.     

On a general class of graph polynomials

Let $\text{F}$ be a family of connected graphs. With each element α ∈ $\text{F}$, we can associate a weight w_α. Let G be a graph. An F-cover of G is a spanning subgraph of G in which every component belongs to $\text{F}$. With every F-cover we can associate a monomial π(C) = Π_αw_α, where the product is taken over all components of the cover. The F-polynomial of G is Σπ(C), where the sum is taken over all F-covers in G. We obtain general results for the complete graph and complete bipartite graphs, and we show that many of the well-known graph polynomials are special cases of more general F-polynomials.     

Universal caterpillars

For a class $\text{C}$ of graphs, denote by u($\text{C}$) the least value of m so that for some graph U on m vertices, every G ? $\text{C}$ occurs as a subgraph of U. In this note we obtain rather sharp bounds on u($\text{C}$) when $\text{C}$ is the class of caterpillars on n vertices, i.e., tree with property that the vertices of degree exceeding one induce a path.     

Proof of a conjecture of S. Chowla

Let θ(k, p) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ … + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0(}\text{mod}\text{p)}$ has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: $\text{θ(k) = O(k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}\text{)}$.     

An approximation algorithm for the hamiltonian walk problem on maximal planar graphs

A hamiltonian walk of a graph is a shortest closed walk that passes through every vertex at least once, and the length is the total number of traversed edges. The hamiltonian walk problem in which one would like to find a hamiltonian walk of a given graph is NP-complete. The problem is a generalized hamiltonian cycle problem and is a special case of the traveling salesman problem. Employing the techniques of divide-and-conquer and augmentation, we present an approximation algorithm for the problem on maximal planar graphs. The algorithm finds, in O(p²) time, a closed spanning walk of a given arbitrary maximal planar graph, and the length of the obtained walk is at most $\text{3}\text{2}$(p ? 3) if the graph has p (≥ 9) vertices. Hence the worst-case bound is $\text{3}\text{2}$.     

Size and independence in triangle-free graphs with maximum degree three

Let C be the class of triangle-free graphs with maximum degree at most three. A lower bound for the number of edges in a graph of C is derived in terms of the number of vertices and the independence. Several classes of graphs for which this bound is attained are given. As corollaries, we obtain the best possible lower bound for the independence ratio of a graph in C and evaluate some Ramsey-type numbers.     

Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem

We study properties of the polynomials φ_k(X) which appear in the formal development Π_{k ? 0}ⁿ (a + bX^k)^rk = Σ_{k ≥ 0}φ_k(X) a^{r ? k}b^k, where r_k ∈ $\text{l}$ and r = Σr_k. this permits us to obtain the coefficients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers G_r(ξ) = Π_{k = 1}^{p ? 1} (a + bξ^k)^kr, where p is a prime, ξ is a primitive pth root of 1, a, b ∈ $\text{l}$ and 1 ≤ r ≤ p ? 3, modulo powers of ξ ? 1 (until (ξ ? 1)^{2(p ? 1) ? r}). This gives more information than the usual logarithmic derivative. Suppose that $\text{p ? ab(a + b)}$. Let $\text{m = ?}\text{b}\text{a}$. We prove that G_r(ξ) ≡ c^p mod p(ξ ? 1)² for some c ∈ $\text{l}$, if and only if Σ_{k = 1}^{p ? 1}k^{p ? 2 ? r}m^k ≡ 0 (mod p). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem.     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

Generating random regular graphs

An algorithm is described which generates a random labeled cubic graph on n vertices. Also described is a procedure which, if successful, generates a random (0,1)-matrix with prescribed row and column sums. The latter yields procedures which, if successful, generate random labeled graphs with specified degree sequence and random labeled bipartite graphs with specified degree sequences. These procedures can be implemented so that each trial requires time which is linear in the number of vertices plus edges, but in generating a random r-regular graph, the probability of success of a given trial is about $\text{exp}\text{(}\text{(1 ? r}{}^2\text{)}\text{4}\text{)}$, which is prohibitively small for large r. Comparisons are made between the complexities of the two methods of generating random cubic graphs. The two general schemes presented derive from methods which have been used to enumerate regular graphs, both asymptotically and exactly.     

On k-sequential and other numbered graphs

The concept of a k-sequential graph is presented as follows. A graph G with ∣V(G)∪ E(G)∣=t is called k-sequential if there is a bijection$\text{?: V(G)∪E(G) → {k,k+1,…,t+k?1}}$ such that for each edge$\text{e}\text{?}\text{=xy}$in E(G) one has$\text{?(}\text{e}\text{?}\text{) = ∣?(x)??(y)∣}$. A graph that is 1-sequential is called simply sequential, and, in particular the author has conjectured that all trees are simply sequential. In this paper an introductory study of k-sequential graphs is made. Further, several variations on the problems of gracefully or sequentially numbering the elements of a graph are discussed.     

Extremal non-bipartite regular graphs of girth 4

Let f(r) denote the smallest number of points in a non-bipartite r-regular graph of girth 4. It is known that $\text{f(r) ≥}\text{5r}\text{2}$ and that $\text{f(r) =}\text{5r}\text{2}$ if r is even. It is proved that $\text{f(r) ～}\text{5r}\text{2}$ and exact values for f(r) are provided for odd integers of the form r = 4n ? 1. Tight bounds for f(r) for odd integers of the form r= 4n + 1 are given.     

Relative lengths of paths and cycles in k-connected graphs

Let G be a k-connected graph where k≥3. It is shown that if G contains a path L of length l then G also contains a cycle of length at least ($\text{(2k ? 4)}\text{(3k ? 4)}$) l. This result is obtained from a constructive proof that G contains 3k² ? 7k + 4 cycles which together cover every edge of L at least 2k² ? 6k + 4 times.     

On some covering designs

Let n ? k ? t be positive integers, and let Ω be a set of n elements. Let C(n, k, t) denote the number of k-tuples of Ω in a minimal system of k-tuples such that every t-tuple is contained in at least one k-tuple of the system. C(n, k, t) has been determined in all cases for which $\text{C(n, k, t) ?}\text{3(t + 1)}\text{2}$ [W. H. Mills, Ars Combinatoria8 (1979), 199–315]. C(n, k, t) is determined in the case $\text{3(t + 1)}\text{2}\text{< C(n, k, t) ?}\text{3(t + 2)}\text{2}$.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and p_ij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff $\text{dim}\text{(x ? y) = s ? 1}$. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p₁₁(x, y) = λ whenever d(x, y) = 1, and p₁₁(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where $\text{2d}\text{3}\text{≤ s < d ? 2}$ and $\text{(s, q, d) ≠ (}\text{2d}\text{3}\text{, 2, d)}$, are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.     

On a function due to S. Chowla

Let θ(k, pⁿ) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ ? + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0}$ (mod pⁿ) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then $\text{θ(k, p}{}^{\mathrm{n}}\text{) ≤ k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n.