Incidence coloring of Mycielskians with fast algorithm

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3.

Extending precolorings to circular colorings

In this paper, we introduce a new variation of list-colorings. For a graph $G$ and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) = \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

Two remarks on the Burr–Erdős conjecture

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log2ψ(G)⌉. In this article, we show that for an interval graph G ⌈log2ψ(G)⌉⩽cub(G)⩽⌈log2ψ(G)⌉+2. It is not clear whether the upper bound of ⌈log2ψ(G)⌉+2 is tight: till now we are unable to find any interval graph with cub(G)>⌈log2ψ(G)⌉. We also show that for an interval graph G, cub(G)⩽⌈log2α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ⌈log2α2⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)⩽cub(G). From the above result, it follows that for any graph G, cub(G)⩽box(G)⌈log2α⌉. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010

Densely connected sub-Gaussian linear structural equation model learning via ℓ1- and ℓ2-regularized regressions

The parameter l(G) for a primitive digraph G introduced by Lewin is the minimum positive integer k for which there are walks of both lengths k and k + 1 from some vertex u to some vertex v. We obtain upper bounds on l(G) if G is primitive ministrong, or G is just primitive and not necessarily ministrong, or G is primitive symmetric. We also discuss the numbers attainable as l(G).

For a clutter \( (C) \) on a finite set E, let \( f(C) \) be the clutter of maximal subsets of E not containing any member of \( (C) \). The loop \( L(C)\) of a clutter C is the finite sequence of clutters \( C,f(C),f^2 (C), \ldots ,f^t (C), \), where t (the length of the loop) is the minimum positive integer with \( f^t (C)\; = \;C. \). Our motivation of the study of the loop lies on the fact that when \( C \) is the set of circuits of a matroid, the loop contains other critical information associated with the matroid, e.g., the set of bases \( f (C) \) and the set of hyperplanes \( f^2 (C). \) We investigate various properties of the loops, in particular, the possible lengths for fixed |E|, the dualities and the symmetries. Our preliminary investigations indicate that there are many interesting problems on the loops whose resolution may provide us with a new insight into Sperner Theory, Matroid Theory and Extremal Set Theory.

Edge-ordered Ramsey numbers

Improved square coloring of planar graphs

Recent experiments have shown ASP solvers to run significantly faster on ground programs of small treewidth. If possible, it may therefore be beneficial to write a non-ground ASP encoding such that grounding it together with an input of small treewidth leads to a propositional program of small treewidth. In this work, we prove that a class of non-ground programs called guarded ASP guarantees this property. Guarded ASP is a subclass of the recently proposed class of connection-guarded ASP, which is known to admit groundings whose treewidth depends on both the treewidth and the maximum degree of the input. Here we show that this dependency on the maximum degree cannot be dropped. Hence, in contrast to connection-guarded ASP, guarded ASP promises good performance even if the input has large maximum degree.

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

It is proved that a planar graph with maximum degree Δ ≥ 11 has total (vertex-edge) chromatic number $Delta; + 1. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 53–59, 1997

Theoretical aspects of equitable partition of networks into sparse modules