Short disjoint cycles in graphs with degree constraints

AbstractWe show that each finite undirected graph G = (V, E), |V| = n, |E|= m with minimum degree δ(G) ⩾ 3 and maximum degree Δ = Δ(G) contains at least n/[4(Δ − 1)log2n] pairwise vertex-disjoint cycles of length at most 4(Δ − 1)log2n. Furthermore collections of such cycles can be determined within O(n(n + m)) steps. For constant Δ this means Ω(n/log n) cycles of length O(log n). This bound is also an optimum.A similar approach yields similar bounds for subgraphs with more edges than vertices instead of cycles. Furthermore also collections of many small pairwise disjoint induced subgraphs of this type can be determined within O(n(n + m)) steps similarly as for cycles

Pushable chromatic number of graphs with degree constraints

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on $\chi_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $\Delta \geq 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $\Delta$ lies between $2^{\frac{\Delta}{2}-1}$ and $(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $\Delta \leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$

Pushable chromatic number of graphs with degree constraints

International audiencePushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as the first general bounds on $\chi_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $\Delta \geq 29$, we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree $\Delta$ lies between $2^{\frac{\Delta}{2}-1}$ and $(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $\Delta \leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$

The number and average size of connected sets in graphs with degree constraints

The average size of connected vertex subsets of a connected graph generalises a much‐studied parameter for subtrees of trees. For trees, the possible values of this parameter are critically affected by the presence or absence of vertices of degree 2. We answer two questions of Andrew Vince regarding the effect of degree constraints on general connected graphs. We give a new lower bound, and the first nontrivial upper bound, on the maximum growth rate of the number of connected sets of a cubic graph, and in fact obtain nontrivial upper bounds for any constant bound on the maximum degree. We show that the average connected set density is bounded away from 1 for graphs with no vertex of degree 2, and generalise a classical result of Jamison for trees by showing that in order for the connected set density to approach 1, the proportion of vertices of degree 2 must approach 1. Finally, we show that any sequence of graphs with minimum degree tending to infinity must have connected set density tending to 1/2

On the utility of network coding in dynamic environments

Many wireless applications, such as ad-hoc networks and sensor networks, require decentralized operation in dynamically varying environments. We consider a distributed randomized network coding approach that enables efficient decentralized operation of multi-source multicast networks. We show that this approach provides substantial benefits over traditional routing methods in dynamically varying environments. We present a set of empirical trials measuring the performance of network coding versus an approximate online Steiner tree routing approach when connections vary dynamically. The results show that network coding achieves superior performance in a significant fraction of our randomly generated network examples. Such dynamic settings represent a substantially broader class of networking problems than previously recognized for which network coding shows promise of significant practical benefits compared to routing