Discrete Mathematics最新文献

Kaneko et al. [12] proved that every 3-connected planar graph G has a 2-connected spanning subgraph K such that $\left|E\right(K\left)\right|\le \frac{4}{3}\left(\right|V\left(G\right)|-1)$, and they also conjectured that the constant of the estimation can be improved to $\frac{4}{3}\left(\right|V\left(G\right)|-2)$ when $\left|V\right(G\left)\right|\ge 8$. To prove the result, they showed the statement for a circuit graph, which is obtained from a 3-connected planar graph by deleting one vertex, and the theorem is best possible for circuit graphs. In this paper, we give a characterization of a circuit graph G each of whose 2-connected spanning subgraph K requires $\left|E\right(K\left)\right|\ge \frac{4}{3}\left(\right|V\left(G\right)|-1)$ and then we improve the bound for the 3-connected planar case.

The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer h, the proper h-conflict-free chromatic number of a graph G, denoted ${\chi }_{\mathrm{pcf}}^h\left(G\right)$, is the minimum k such that G has a proper k-coloring where every vertex v has $\min \{{\mathrm{deg}}_G(v),h\}$ colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if G is a graph with $\Delta \left(G\right)\ge 3$, then ${\chi }_{\mathrm{pcf}}^1\left(G\right)\le \Delta \left(G\right)+1$. The best known result regarding the conjecture is ${\chi }_{\mathrm{pcf}}^1\left(G\right)\le 2\Delta \left(G\right)+1$, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all h, and also enlarge the class of graphs for which the conjecture is known to be true.

Our main result is the following: for a graph G, if $\Delta \left(G\right)\ge h+2$, then ${\chi }_{\mathrm{pcf}}^h\left(G\right)\le (h+1)\Delta \left(G\right)-1$; this is tight up to the additive term as we explicitly construct infinitely many graphs G with ${\chi }_{\mathrm{pcf}}^h\left(G\right)=(h+1)\left(\Delta \right(G)-1)$. We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for h-dynamic coloring.

This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph t-clean if it does not contain as an induced subgraph the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, subdivisions of a $(t\times t)$-wall, and line graphs of subdivisions of a $(t\times t)$-wall. It is known that graphs with bounded treewidth must be t-clean for some t; however, it is not true that every t-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of ($IS{K}_4$, wheel)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that t-clean ($IS{K}_4$, wheel)-free graphs have bounded treewidth and that t-clean graphs with no cycle with a unique chord have bounded treewidth.

Extending a proposal of Defant and Kravitz (2024) [2], we define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems give optimal residual information regarding the Hitomezashi loop length, loop count, as well as possible homology classes of such loops. Special attention is paid to toroidal Hitomezashi patterns that are symmetric with respect to the diagonal $x=y$, where we establish a novel connection between Hitomezashi and knot theory.

The double star $S(m_1,m_2)$ is obtained from joining the centres of a star with $m_1$ leaves and a star with $m_2\le m_1$ leaves. We give a short proof of a new upper bound on the two-colour Ramsey number of $S(m_1,m_2)$ which holds for all $m_1,m_2$ with $\frac{\sqrt{5}+1}{2}{m}_2<m_1<3m_2$. Our result implies that for all positive m, the Ramsey number of the double star $S(2m,m)$ is at most $\lceil 4.275m\rceil +1$.

In this paper, we construct a class of linear complementary dual (LCD for short) 2-quasi-abelian codes over a finite field. Based on counting the number of such codes and estimating the number of the codes in this class whose relative minimum weights are small, we prove that the class of LCD 2-quasi-abelian codes over any finite field is asymptotically good. The existence of such codes is unconditional, which is different from the case of self-dual 2-quasi-abelian codes over a special finite field.

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of G is the minimum k such that G has a 2-distance k-coloring, denoted by ${\chi }_2\left(G\right)$. In this paper, we show that ${\chi }_2\left(G\right)\le 17$ for every planar graph G with maximum degree $\Delta \left(G\right)\le 5$, which improves a former bound ${\chi }_2\left(G\right)\le 18$.

An edge-coloring of a graph G with colors $1,\dots ,t$ is called an interval t-coloring if all colors are used and the colors of edges incident to each vertex of G are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph G with at least one edge has an interval t-coloring, then $t\le 2\left|V\right(G\left)\right|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph G admits an interval t-coloring, then $t\le \frac{11}{6}\left|V\right(G\left)\right|$. In the same paper Axenovich suggested a conjecture that if a planar graph G has an interval t-coloring, then $t\le \frac{3}{2}\left|V\right(G\left)\right|$. In this paper we first prove that if a graph G has an interval t-coloring, then $t\le \frac{\left|E\right(G\left)\right|+\left|V\right(G\left)\right|-1}{2}$. Next, we confirm Axenovich's conjecture by showing that if a planar graph G admits an interval t-coloring, then $t\le \frac{3\left|V\right(G\left)\right|-4}{2}$. We also prove that if an outerplanar graph G has an interval t-coloring, then $t\le \left|V\right(G\left)\right|-1$. Moreover, all these upper bounds are sharp.

A non-zero $F$-linear map from a finite-dimensional commutative $F$-algebra to the field $F$ is called an $F$-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an $F_2$-valued trace of the $F_2$-algebra $R_2:=F_2\left[x\right]/〈x^3-x〉$ to study binary subfield code $C_D^{\left(2\right)}$ of $C_D:=\{{(x\cdot d)}_{d\in D}:x\in R_2^m\}$ for each defining set D derived from a certain simplicial complex. For $m\in N$ and $X\subseteq \{1,2,\dots ,m\}$, define ${\Delta }_X:=\{v\in F_2^m:\text{Supp}(v)\subseteq X\}$ and $D:=(1+u^2)D_1+u^2{D}_2+(u+u^2)D_3$, a subset of $R_2^m$, where $u=x+〈x^3-x〉,D_1\in \{{\Delta }_L,{\Delta }_L^{}$