A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let $G$ be a balanced bipartite graph of order $2n\ge 4$, and let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. In 1963, Moon and Moser proved that if ${\sigma }_{1,1}\left(G\right)\ge n+1$, then $G$ is hamiltonian. In this note, we show that if $k$ is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly $k$ cycles for sufficiently large graphs. In order to prove this, we also give a ${\sigma }_{1,1}$ condition for the existence of $k$ vertex-disjoint alternating cycles with respect to a chosen perfect matching in $G$.     

The confirmation of a conjecture on disjoint cycles in a graph

Let $t$ and $k$ be two integers with $t\ge 5$ and $k\ge 2$. For a graph $G$ and a vertex $x$ of $G$, we use $d_G\left(x\right)$ to denote the degree of $x$ in $G$. Define ${\sigma }_t\left(G\right)$ to be the minimum value of ${\sum }_{x\in X}{d}_G\left(x\right)$, where $X$ is an independent set of $G$ with $\left|X\right|=t$. This paper proves the following conjecture proposed by Gould et al. (2018). If $G$ is a graph of sufficiently large order with ${\sigma }_t\left(G\right)\ge 2kt?t+1$, then $G$ contains $k$ vertex-disjoint cycles.     

Disjoint cycles with chords in graphs

Let be integers, , , and let for each , be a cycle or a tree on vertices. We prove that every graph G of order at least n with contains k vertex disjoint subgraphs , where , if is a tree, and is a cycle with chords incident with a common vertex, if is a cycle. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 87–98, 2009     

A note on the strength and minimum color sum of bipartite graphs

The strength of a graph G is the smallest integer s such that there exists a minimum sum coloring of G using integers {1,…,s}, only. For bipartite graphs of maximum degree Δ we show the following simple bound: s≤⌈Δ/2⌉+1. As a consequence, there exists a quadratic time algorithm for determining the strength and minimum color sum of bipartite graphs of maximum degree Δ≤4.     

Alternating hamiltonian cycles in two colored complete bipartite graphs

We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors.     

Large bipartite Cayley graphs of given degree and diameter

Let BC_d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC_d,k≥2(k−1)((d−4)/3)^k−1 for any d≥6 and any even k≥4, and BC_d,k≥(k−1)((d−2)/3)^k−1 for d≥6 and k≥7 such that k≡3 (mod 4).     

Fan-type condition on disjoint cycles in a graph

Let $k$ be a positive integer. In this paper, we prove that for a graph $G$ with at least $4k$ vertices, if max$\{d\left(x\right),d\left(y\right)\}\ge 2k$ for any pair of nonadjacent vertices $\{x,y\}?V\left(G\right)$, then $G$ contains $k$ disjoint cycles. This generalizes the results given by Corrá di and Hajnal (1963), Enomoto (1998), and Wang (1999).     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

Extreme values of the sum of squares of degrees of bipartite graphs

In this paper we determine the minimum and maximum values of the sum of squares of degrees of bipartite graphs with a given number of vertices and edges.     

Independent dominating sets and hamiltonian cycles

A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r‐regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 233–244, 2007     

Eulerian straight ahead cycles in drawings of complete bipartite graphs

Straight ahead cycles in drawings of graphs pass all vertices leaving the same number of edges on each side. For complete bipartite graphs K_2r,2s, drawings with straight ahead Eulerian cycles and with straight ahead 4-cycles are constructed.     

Disjoint paths in tournaments

Given k   pairs of vertices (_si,_ti)$(s_i,t_i)$(1≤i≤k)$(1\le i\le k)$ of a digraph G, how can we test whether there exist k   vertex-disjoint directed paths from _si$s_i$ to _ti$t_i$ for 1≤i≤k$1\le i\le k$? This is NP-complete in general digraphs, even for k=2$k=2$ [2], but for k=2$k=2$ there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1]. Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete.