An ore-type condition for the existence ofk-factors in graphs

Partitions of graphs and multigraphs under degree constraints

For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ exists such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every pair of distinct vertices $u,v$ at distance at most $r$ in $G$. It was conjectured that $\chi'_{\Sigma,r}(G)\leq (1+o(1))\Delta^{r-1}$ for every $r\geq 3$. Thus far it has been in particular proved that $\chi'_{\Sigma,r}(G)\leq 6\Delta^{r-1}$ if $r\geq 4$. Combining probabilistic and constructive approach, we show that this can be improved to $\chi'_{\Sigma,r}(G)\leq (4+o(1))\Delta^{r-1}$ if the minimum degree of $G$ equals at least $\ln^8\Delta$.

On independent generalized degrees and independence numbers in K(1, m)-free graphs

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

3-component domination numbers in graphs

Given a simple graph G, m a positive integer. The square of path graph P_m, denoted by P_m^2, is a graph obtained from P_m by adding new edges between any pair of vertices at distance at most 2 in P_m. A graph G is P_m^2-saturated if G does not contain P_m^2 as a subgraph, but the addition of any edge between two nonadjacent vertices in G contain P_m^2. The minimum size of P_m^2-saturated graph on n vertices is called a saturation number for P_m^2, denoted by sat(n,P_m^2). A set Sat(n,P_m^2 )={G:|V(G)|=sat(n,P_m^2) and G a P_m^2-saturated graph}. All graphs in Sat(n,P_m^2) are obtained computationally for n≤8 and m≤8 and expressed by their degree sequence.

Spectra of quasi-strongly regular graphs

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$n?3 and W a subset of V(G) with $$|W|\ge 3k$$|W|?3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)?2n/3 for each $$x\in W$$x?W, then G contains k vertex-disjoint cycles such that each of them contains at least three vertices of W. In this paper, we obtain an analogue result of Wang's Theorem in bipartite graph with the partial degree condition. Let $$G=(V_1,V_2;E)$$G=(V1,V2?E) be a bipartite graph with $$|V_1|=|V_2|=n$$|V1|=|V2|=n, and let W be a subset of $$V_1$$V1 with $$|W|\ge 2k$$|W|?2k, where k is a positive integer. We show that if $$d(x)+d(y)\ge n+k$$d(x)+d(y)?n+k for every pair of nonadjacent vertices $$x\in W, y\in V_2$$x?W,y?V2, then G contains k vertex-disjoint cycles such that each of them contains at least two vertices of W.

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ⩾ 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ⩾ 2δ2, then for any 3 ⩽ l1 ⩽ l2 ⩽ ⃛ ⩽ lk ⩽ n, 1 = k = [(δ ‐ 1)/2], such graph has K edge disjoint cycles with lengths l1, l2…lk, respectively. In particular, when l1 = l2 = ⃛ = lk = n and k = [(δ ‐ 1)/2], the graph contains [(δ ‐ 1)/2] edge disjoint hamiltonian cycles.

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -- which asks for a subgraph with minimum degree that connects a given set of vertices -- is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.The standard techniques for solving degree-bounded problems often fall in the category of iterative and dependent rounding techniques. Unfortunately, these rounding methods are inherently difficult to adapt to an online settings since the underlying fractional solution may change dramatically in between the rounding steps. Indeed, this might be the very reason that despite many advances in the online network design paradigm in the past two decades, the natural family of degree-bounded problems has remained widely open.In this paper, we design an intuitive greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Furer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach and its dual analysis, paves the way for solving the online variants of the classical problems in this family of problems.

formula omitted]-factors in graphs

For two graphs F, H and a positive integer n, the function fF,H(n) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where F is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer r and every Kr-1-free graph F, there exists some εF>00$$\\end{document}]]> such that fF,Kr(n)=O(n1/2-εF). This result is tight in two ways. Firstly, it is no longer true if F contains Kr-1 as a subgraph. Secondly, we show that for all r≥4 and ε>00$$\\end{document}]]>, there exists a Kr-1-free graph F for which fF,Kr(n)=Ω(n1/2-ε). Along the way of proving this, we show in particular that for every graph F with minimum degree t, we have fF,K4(n)=Ω(n1/2-6/t). This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants 0<c<C such that for each r≥4, if F is a bipartite graph with sufficiently large minimum degree, then Ω(nclogr)≤fF,Kr(n)≤O(nClogr). This shows that for graphs F with large minimum degree, the behaviour of fF,Kr(n) is drastically different from that of the corresponding off-diagonal Ramsey number fK2,Kr(n).

On a Hamiltonian Cycle in Which Specified Vertices Are Uniformly Distributed

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n − 1)k + 4k(n − 4) + 2·3k − 2((δ + 1)k − δk)(3n − 6 − m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0⩽3n − 6 − m≤⌊n/2⌋ − 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112-123, 2011 © 2011 Wiley Periodicals, Inc.

Two remarks on the Burr–Erdős conjecture