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A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set. In this paper, we study the following conjecture posed by Goddard and Henning (Discrete Math. 313:839–854, 2013): If G ≇ K 3 , 3 is a connected, cubic, bipartite graph on n vertices, then i (G) ≤ 4 11 n . Henning et al. (Discrete Appl. Math. 162:399–403, 2014) prove the conjecture when the girth is at least 6. In this paper we strengthen this result by proving the conjecture when the graph has no subgraph isomorphic to K 2 , 3 . [ABSTRACT FROM AUTHOR]

Let T be a tree with m edges. It was conjectured that every m-regular bipartite graph can be decomposed into edge-disjoint copies of T. In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on n vertices can be decomposed into n 2 paths, which is related to the well-known Gallai's Conjecture: every connected graph on n vertices can be decomposed into at most n + 1 2 paths. [ABSTRACT FROM AUTHOR]

A graph is called $$2K_2$$ -free if it does not contain two independent edges as an induced subgraph. Gao and Pasechnik conjectured that every $$\frac{3}{2}$$ -tough $$2K_2$$ -free graph with at least three vertices has a spanning trail with maximum degree at most 4. In this paper, we confirm this conjecture. We also provide examples for all $$t < \frac{5}{4}$$ of t-tough graphs that do not have a spanning trail with maximum degree at most 4. [ABSTRACT FROM AUTHOR]

The semitotal domination number of a graph G without isolated vertices is the minimum cardinality of a set S of vertices of G such that every vertex in $$V(G){\setminus } S$$ is adjacent to at least one vertex in S, and every vertex in S is within distance 2 of another vertex of S. In Henning and Marcon (Ann Comb 20(4):1-15, 2016), it was shown that every connected claw-free cubic graph G of order n has semitotal domination number at most $$\frac{4n}{11}$$ when $$n\ge 10$$ , and it was conjectured that the bound can be improved from $$\frac{4n}{11}$$ to $$\frac{n}{3}$$ if $$G\notin \{K_4,N_2\}$$ . In this paper, we prove this conjecture. [ABSTRACT FROM AUTHOR]

A graphic sequence π = ( d, d, . . . , d) is said to be potentially K-graphic if there is a realization of π containing K as a subgraph, where K is the 1 × 1 × s complete 3-partite graph. In this paper, a simple characterization of potentially K-graphic sequences for s ≥ 2 and n ≥ 3 s + 1 is obtained. This characterization implies Lai's conjecture on σ( K, n), which was confirmed by J.H. Yin, J.S. Li and W.Y. Li, and the values of σ( K, n) for s ≥ 4 and n ≥ 3 s + 1, where K is the 2 × s complete bipartite graph. [ABSTRACT FROM AUTHOR]

We consider the connected graphs G that satisfy the following property: If $$n \gg m \gg k$$ are integers, then any coloring of the edges of $$K_{n}$$ , using m colors, containing no properly colored copy of G, contains a monochromatic k-connected subgraph of order at least $$n - f(G, k, m)$$ where f does not depend on n. If we let $$\mathscr {G}$$ denote the set of graphs satisfying this statement, we exhibit some infinite families of graphs in $$\mathscr {G}$$ as well as conjecture that the cycles in $$\mathscr {G}$$ are precisely those whose lengths are divisible by 3. Our main result is that $$C_{6} \in \mathscr {G}$$ . [ABSTRACT FROM AUTHOR]

A graph G is prime if the vertices can be distinctly labeled with the integers 1,2,...,|V(G)| so that adjacent vertices have relatively prime labels. We show that every cubic bipartite graph is prime except K3,3, which implies a number of other results. We also provide evidence to support a conjectured classification for the primality of 2-regular graphs. [ABSTRACT FROM AUTHOR]

There is a sufficiently large N∈hN such that the following holds. If G is a tripartite graph with N vertices in each vertex class such that every vertex is adjacent to at least 2N/3+2h-1 vertices in each of the other classes, then G can be tiled perfectly by copies of Kh,h,h. This extends work by Martin and Zhao (Electron J Combin 16(1):109, 2009) and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that 2N/3+2h-1 in our result can not be replaced by 2N/3+h-2 and that if N is divisible by 6h, then we can replace it with the value 2N/3+h-1 and this is tight. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of chromatic number k. For a k-coloring f of G, a full f-rainbow path is a path of order k in G whose vertices are all colored differently by f. We show that G has a k-coloring f such that every vertex of G lies on a full f-rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper vertex-colorings, Graphs Combin. 23 2007, 201-203). Furthermore, we show that if G has a cycle of length 0 modulo k, then G has a k-coloring f such that, for every vertex u of G, some full f-rainbow path begins at u, which solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic graphs, ) and verifies some special cases of a conjecture of Akbari et al. (Colorful paths in vertex coloring of graphs, preprint). Finally, we establish some more results on the existence of optimal colorings with (directed) full rainbow paths. [ABSTRACT FROM AUTHOR]