Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs

Let ?1(G) and q1(G) be the spectral radius and the signless Laplacian spectral radius of a kuniform hypergraph G, respectively. In this paper, we give the lower bounds of d-?1(H) and 2d-q1(H), where H is a proper subgraph of a f (-edge)-connected d-regular (linear) k-uniform hypergraph. Meanwhile, we also give the lower bounds of 2?-q1(G) and ?-?1(G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ?.

Kempe-Locking Configurations

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

The Thickness of the Cartesian Product of Two Graphs

The thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.

(a,d)-Continuous monotonic subgraph decomposition of K_n+1 and integral sum graphs G_(0,n)

For $a,d,n \in \mathbb{N}$, we define$(a, d)-Continuous$ Monotonic Subgraph Decomposition or $(a,d)-CMSD$ of a graph $G$ of size $\frac{(2a+(n-1)d)n}{2}$ as the decomposition of $G$ into $n$ subgraphs $G_1,G_2,\ldots,G_n$ without isolated vertices such that each $G_i$ is connected and isomorphic to a proper subgraph of $G_{i+1}$ and $|E(G_i)| = a+(i-1)d$ for $i = 1,2,\ldots,n.$ $(1, 1)-CMSD$ of a graph $G$ is called a Continuous Monotonic Subgraph Decomposition or CMSD of $G$. Harary introduced the concepts of sum and integral sum graphs and a family of integral sum graphs $G_{-n,n}$ over $[-n, n]$ and it was generalized to $G_{-m,n}$ where $[r, s] = \{r,r+1,\ldots,s\}$, $r,s \in \mathbb{Z}$ and $m,n \in \mathbb{N}_0$. In this paper, we study $(a, d)-CMSD$ of $K_{n+1}$ and $G_{0,n}$ into families of triangular books, triangular books with book mark and Fans with handle.

Defective Choosability of Graphs without Small Minors

For each proper subgraph $H$ of $K_5$, we determine all pairs $(k,d)$ such that every $H$-minor-free graph is $(k,d)^*$-choosable or $(k,d)^-$-choosable. The main structural lemma is that the only 3-connected $(K_5-e)$-minor-free graphs are wheels, the triangular prism, and $K_{3,3}$; this is used to prove that every $(K_5-e)$-minor-free graph is 4-choosable and $(3,1)$-choosable.

The Spectral Radius of Subgraphs of Regular Graphs

Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ be the largest and smallest eigenvalues of the adjacency matrix of a graph $G$. Our main results are: (i) Let $G$ be a regular graph of order $n$ and finite diameter $D.$ If $H$ is a proper subgraph of $G,$ then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over nD}. $$ (ii) If $G$ is a regular nonbipartite graph of order $n$ and finite diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{1\over nD}. $$

Revisiting Two Classical Results on Graph Spectra

Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ be the largest and smallest eigenvalues of the adjacency matrix of a graph $G$. Our main results are: (i) If $H$ is a proper subgraph of a connected graph $G$ of order $n$ and diameter $D$, then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over\mu\left( G\right) ^{2D}n}. $$ (ii) If $G$ is a connected nonbipartite graph of order $n$ and diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{2\over\mu\left( G\right) ^{2D}n}. $$ For large $\mu $ and $D$ these bounds are close to the best possible ones.

Ramsey ($K_{1,2},K_3$)-Minimal Graphs

For graphs $G,F$ and $H$ we write $G\rightarrow (F,H)$ to mean that if the edges of $G$ are coloured with two colours, say red and blue, then the red subgraph contains a copy of $F$ or the blue subgraph contains a copy of $H$. The graph $G$ is $(F,H)$-minimal (Ramsey-minimal) if $G\rightarrow (F,H)$ but $G'\not\rightarrow (F,H)$ for any proper subgraph $G'\subseteq G$. The class of all $(F,H)$-minimal graphs shall be denoted by $R (F,H)$. In this paper we will determine the graphs in $R(K_{1,2},K_3)$.