For a group \(T\) and a subset \(S\) of \(T\), the bi-Cayley graph \(\text{BCay}(T, S)\) of \(T\) with respect to \(S\) is the bipartite graph with vertex set \(T \times \{0, 1\}\) and edge set \(\{\{(g, 0), (ag, 1)\} | g \in T, s \in S\}\). In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infinite families of cubic semisymmetric graphs.

We study the notion of path-congruence \(\Phi: T_1 \rightarrow T_2\) between two trees \(T_1\) and \(T_2\). We introduce the concept of the trunk of a tree, and prove that, for any tree \(T\), the trunk and the periphery of \(T\) are stable. We then give conditions for which the center of \(T\) is stable. One such condition is that the central vertices have degree \(2\). Also, the center is stable when the diameter of \(T\) is less than \(8\).

We call a cycle whose length is at most \(5\) a short cycle. In this paper, we consider the packing of short cycles in a graph with specified edges. A minimum degree condition is obtained, which is slightly weaker than that of the result in \([1]\).

Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \(f\)-factor if \(d_G^{h}(x) = f(x)\) for every \(x \in V(F)\). In this paper, we prove that if \(\delta(G) \geq b\) and \(\alpha(G) \leq \frac{4a(\delta-b)}{(b+1)^2}\), then \(G\) has a fractional \(f\)-factor. Where \(a\) and \(b\) are integers such that \(0 \leq a \leq f(x) \leq b\) for every \(x \in V(G)\). Therefore, we prove that the fractional analogue of Conjecture in \([2]\) is true.

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group, \(g \in A\) and \(\Gamma\) a group of automorphisms of \(D\). We consider the number of \(T\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. Specifically, we enumerate the number of \(I\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order and the trivial automorphism group \(\Gamma\) of \(D\), when \(A\) is the cyclic group \({Z}_{p^n}\) and the direct sum of \(m\) copies of \({Z}_p\) for any prime number \(p (> 2)\).

The Grundy number of an impartial game \(G\) is the size of the unique Nim heap equal to \(G\). We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these “fractal sequences” and give a bijection between these sequences and certain upper-triangular arrays. As a special case, we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap.

A graph \(G\) is clique-perfect if the cardinality of a maximum clique-independent set of \(H\) is equal to the cardinality of a minimum clique-transversal of \(H\), for every induced subgraph \(H\) of \(G\). When equality holds for every clique subgraph of \(G\), the graph is \(c\)-clique-perfect. A graph \(G\) is \(K\)-perfect when its clique graph \(K(G)\) is perfect. In this work, relations are described among the classes of perfect, \(K\)-perfect, clique-perfect and \(c\)-clique-perfect graphs. Besides, partial characterizations of \(K\)-perfect graphs using polyhedral theory and clique subgraphs are formulated.

In this note, we investigate arithmetic properties of the Motzkin numbers. We prove that for large \(n\), the product of the first \(n\) Motzkin numbers is divisible by a large prime. The proofs use the Deep Subspace Theorem.

The point-distinguishing chromatic index of a graph \(G\), denoted by \(\chi_o(G)\), is the smallest number of colours in a (not necessarily proper) edge colouring of \(G\) such that any two distinct vertices of \(G\) are distinguished by sets of colours of their adjacent edges. The exact value of \(\chi_o(K_{m,n})\) is found if either \(m \leq 10\) or \(n \geq 8m^2 – 2m + 1\).

Star graphs were introduced by \([1]\) as a competitive model to the \(n\)-cubes. Then hyper-stars were introduced in \([9]\) to be a competitive model to both \(n\)-cubes and star graphs. In this paper, we discuss strong connectivity properties and orientability of the hyper-stars.