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This article deals with homomorphisms of oriented graphs with respect to push equivalence. Here homomorphisms refer to arc preserving vertex mappings, and push equivalence refers to the equivalence class of orientations of a graph $G$ those can be obtained from one another by reversing arcs of an edge cut. We study and prove some fundamental properties of pushable homomorphisms, and establish its connections to homomorphisms of signed graphs and graph coloring. To list a few highlights of this work: $\bullet$ We characterize orientations of a graph up to push equivalence and show that it is possible to decide whether they are equivalent or not in polynomial time. $\bullet$ We give a canonical definition of pushable homomorphism - this answers a natural open question. $\bullet$ We build a one-to-one correspondence between the equivalence classes of oriented and signed bipartite graphs. Thus, it is possible to translate a number of important results directly from the theory of signed graphs to oriented graphs. In particular, we show that pushable homomorphisms of bipartite graphs capture the entire theory of graph coloring as a subcase. $\bullet$ Given a graph $G$, we build a gadget oriented graph $\overrightarrow{G}^{(k)}$ which admits a pushable homomorphism to a directed odd cycle of length $(2k+1)$ if and only if $G$ admits a $(2k+1)$-coloring. We also show that it is NP-complete to determine whether an oriented (sparse) graph admits a pushable homomorphism to a directed odd cycle or not.

An oriented graph has weak diameter at most $d$ if every non-adjacent pair of vertices are connected by a directed $d$-path. The function $f_d(n)$ denotes the minimum number of arcs in an oriented graph on $n$ vertices having weak diameter $d$. Finding the exact value of $f_d(n)$ is a challenging problem even for $d = 2$. This function was introduced by Katona and Szemeredi (1967), and after that several attempts were made to find its exact value by Znam (1970), Dawes and Meijer (1987), Furedi, Horak, Pareek and Zhu (1998), and Kostochka, Luczak, Simonyi and Sopena (1999) through improving its best known bounds. In that process, it was proved that this function is asymptotically equal to $n\log_2 n$ and hence, is an asymptotically increasing function. However, the exact value and behaviour of this function was not known. In this article, we observe that the oriented graphs with weak diameter at most $2$ are precisely the absolute oriented cliques, that is, analogues of cliques for oriented graphs in the context of oriented coloring. Through studying arc-minimal absolute oriented cliques we prove that $f_2(n)$ is a strictly increasing function. Furthermore, we improve the best known upper bound of $f_2(n)$ and conjecture that our upper bound is tight. This improvement of the upper bound improves known bounds involving the oriented achromatic number.

In this work we consider arc criticality in colourings of oriented graphs. We study deeply critical oriented graphs, those graphs for which the removal of any arc results in a decrease of the oriented chromatic number by $2$. We prove the existence of deeply critical oriented cliques of every odd order $n\geq 9$, closing an open question posed by Borodin et al. (Journal of Combinatorial Theory, Series B, 81(1):150-155, 2001). Additionally, we prove the non-existence of deeply critical oriented cliques among the family of circulant oriented cliques of even order., Comment: 9 pages, 2 figures