Orientations of graphs avoiding given lists on out‐degrees

Abstract

Let G be a graph and F : V ( G ) → 2 N be a mapping. The graph G is said to be F‐ avoiding if there exists an orientation O of G such that d O + ( v ) ∉ F ( v ) for every v ∈ V ( G ), where d O + ( v ) denotes the out‐degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and ∣ F ( v ) ∣ ≤ d G ( v ) / 2 for every v ∈ V ( G ), then G is F‐avoiding. The bound ∣ F ( v ) ∣ ≤ d G ( v ) / 2 is best possible. For every graph G, we conjecture that if ∣ F ( v ) ∣ ≤ 1 2 ( d G ( v ) − 1 ) for every v ∈ V ( G ), then G is F‐avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.

The existence of $\{p,q\}$-orientations in edge-connected graphs

“…In this subsection, we are going to prove the following assertion on the existence of orientations with sparse lists on out-degrees in partition-connected graphs. For dense lists in all graphs, it was investigated by Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati (2020) [3]. Before stating the main result, we need to recall the following lemma from [7].…”

List-avoiding orientations

“…One tool that was used extensively in [1] is the following theorem, called the Combinatorial Nullstellensatz, introduced by Alon and Tarsi in [3] and further developed as a tool by Alon [2].…”

“…An orientation of a graph G is an assignment of a direction uv or vu to each edge {u, v} ∈ E(G). For an orientation D of a graph G and a vertex v ∈ V (G), we denote by E + (v) the arcs outgoing from v in D, and we denote by E − (v) the arcs incoming to v. We write deg Recently, Akbari, Dalirrooyfard, Ehsani, Ozeki and Sherkati [1] considered the similar problem of finding an orientation of a graph that avoids a certain out-degree at each vertex. Given a graph G and a function f : V (G) − → N, we say that an orientation D of G is f -avoiding if deg + D (v) = f (v) for each v ∈ V (G).…”

“…Given a graph G and a function f : V (G) − → N, we say that an orientation D of G is f -avoiding if deg + D (v) = f (v) for each v ∈ V (G). It was proved in [1] that there is an f -avoiding orientation for every 2-connected graph G that is not an odd cycle and for every function f : V (G) − → N, and that an odd cycle has an f -avoiding orientation if and only if f (v) = 1 for some vertex v of the cycle. Frank, Tardos, and Sebő [11] considered the same problem modulo 2, or in other words, with parity constraints on the out-degrees of the vertices in G.…”

A necessary and sufficient condition for the existence of $\{p,p+1,q-1,q\}$-orientations in simple graphs

“…Remark 3.4. Note that one can use Lemma 3.2 to rediscover Theorem 4 and Lemma 6 in [3], which gave sufficient conditions for the existence of list orientations, from Theorem 1 in [5] and Theorem 2 in [5], which gave sufficient conditions for the existence of list factors.…”