NZ‐flows in strong products of graphs

Abstract

Abstract:We prove that the strong product G 1 G 2 of G 1 and G 2 is Z 3 -flow contractible if and only if G 1 G 2 is not T K 2 , where T is a tree (we call T K 2 a K 4 -tree). It follows that G 1 G 2 admits an NZ 3-flow unless G 1 G 2 is a K 4 -tree. We also give a constructive proof that yields a polynomial algorithm whose output is an NZ 3-flow if G 1 G 2 is not a K 4 -tree, and an NZ 4-flow otherwise. ᭧

Flows on the join of two graphs

“…The flow number of the direct product of graphs was determined by Zhang, Zheng and Mamut [20]. Imrich, Peterin, Špacaran and Zhang [5] considered the flow number of the strong product of two graphs.…”

“…The 3-flow conjecture has received somewhat less attention. Perhaps the lack of progress in this area has motivated several authors to examine flow numbers of graphs resulting from various graph operations, most notably graph products [5], [6], [10], [12], [15], [20]. Very recently Thomassen proved the 3-flow conjecture for 8-edge-connected graphs [17].…”

Integer flows on triangularly connected signed graphs

“…A graph is locally connected if the subgraph induced by the neighbor of each vertex is connected. It is known that locally connected graphs, square of graphs, chordal graphs, triangulations on surfaces, and some types of products of graphs are triangularly connected (such as [8, 10], for ordinary graphs) and thus we have the following corollary.…”

Nowhere-zero 3-flows in semistrong product of graphs

“…was investigated by Zhang et al [17] and Imrich et al [4], respectively. More results on nowhere-zero 3-flows and group connectivity of products of graphs can be found in [9,15,16] and others.…”