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Henning et al. (Discrete Appl Math 162:399–403, 2014) proved that if G is a bipartite, cubic graph of order n and of girth at least 6, then i (G) ≤ 4 11 n . In this paper, we improve the 4 11 -bound to a 5 14 -bound, and prove that if G is a bipartite, cubic graph of order n and of girth at least 6, then i (G) ≤ 5 14 n . [ABSTRACT FROM AUTHOR]

For a graph G, define σ 2 (G) by σ 2 (G) = min { d G (x) + d G (y) : x , y ∈ V (G) , x ≠ y , x y ∉ E (G) } . If G is a bipartite graph with partite sets X and Y, we also define σ 1 , 1 (G) by σ 1 , 1 (G) = min { d G (x) + d G (y) : x ∈ X , y ∈ Y , x y ∉ E (G) } . Ore's theorem states that a graph of order n ≥ 3 with σ 2 (G) ≥ n contains a hamiltonian cycle and the Moon–Moser theorem states that a balanced bipartite graph G of order 2 n ≥ 4 with σ 1 , 1 (G) ≥ n + 1 contains a hamiltonian cycle. In Chen et al. (Discrete Math 343:Article No. 111663, 2020), we studied the relationship between Ore's theorem and the Moon–Moser theorem, and proved that the refinement of the Moon–Moser theorem given by Ferrara et al. (Discrete Math 312:459–461, 2012) implies Ore's theorem for graphs of even order. In this paper, we extend the above study to the graphs of odd order. Since no graphs of odd order contain a spanning balanced bipartite subgraph, the Moon–Moser theorem does not work in this case. We instead introduce its counterpart for the graphs in which the orders of the partite sets differ by 1, proved in Matsubara et al. (Discrete Math 340:87–95, 2017). We refine this result and prove that this refinement implies Ore's theorem. [ABSTRACT FROM AUTHOR]

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]

A total weighting of a graph G is a mapping $$\varphi $$ that assigns a weight to each vertex and each edge of G. The vertex-sum of $$v \in V(G)$$ with respect to $$\varphi $$ is $$S_{\varphi }(v)=\sum _{e\in E(v)}\varphi (e)+\varphi (v)$$ . A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph $$G=(V,E)$$ is called $$(k,k{^{\prime }})$$ -choosable if the following is true: For any total list assignment L which assigns to each vertex v a set L( v) of k real numbers, and assigns to each edge e a set L( e) of $$k{^{\prime }}$$ real numbers, there is a proper total weighting $$\varphi $$ with $$\varphi (y)\in L(y)$$ for any $$y \in V \cup E$$ . In this paper, we prove that for any graph $$G\ne K_1$$ , for any positive integer m, the m-cone graph of G is (1, 4)-choosable. Moreover, we give some sufficient conditions for the m-cone graph of G to be (1, 3)-choosable. In particular, if G is a tree, a complete bipartite graph or a generalized $$\theta $$ -graph, then the m-cone graph of G is (1, 3)-choosable. [ABSTRACT FROM AUTHOR]

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99-107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n - 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G. [ABSTRACT FROM AUTHOR]

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ i = |{ x | ∂( u, x) = ∂( v, x) = 1 and ∂( w, x) = i − 1}| ( i = 2, 3,..., d) depends only on i whenever ∂( u, v) = 2 and ∂( u, w) = ∂( v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph. [ABSTRACT FROM AUTHOR]

In a complete bipartite decomposition π of a graph, we consider the number ϑ( v;π) of complete bipartite subgraphs incident with a vertex v. Let ϑ( G)= $$\min \limits_{\pi } \max \limits_{v\in V(G)}$$ ϑ( v;π). In this paper the exact values of ϑ( G) for complete graphs and hypercubes and a sharp upper bound on ϑ( G) for planar graphs are provided, respectively. An open problem proposed by P.C. Fishburn and P.L. Hammer is solved as well. [ABSTRACT FROM AUTHOR]